Before I launch into a more full response here, I just want to share one story. In the winter of 1835, there was an attempt to assassinate President Andrew Jackson, the seventh president of the United States. This was the first assassination attempt on the life of a President. It's said that the would-be assassin produced one pistol, which misfired, followed by another, which also misfired, while trying to hit the president from only a yard and a half away (4.5 feet)! (Tucker, 195)^^1

Richard Henry Wilde, a poet from Georgia and a Representative in the House at the time, wrote in a letter that same day that the President attempted to rush the would-be assassin with his umbrella or stick (confusion persists), and that everyone from Congressmen to Naval Assistants tried to rush the attempted murderer, Richard Lawrence. (Tucker, 198)

This alone speaks to the security surrounding the President at the time. Though this occurred in 1835, it's often noted that the "Secret Service" was not created until 1860 by the Department of the Treasury, meant to investigate counterfeiting operations essentially. This was the official "United States Secret Service". At the time, there was actually another organization, called the National Detective Bureau, that was also frequently referred to as the Secret Service, and they were more focused on protecting Lincoln and other wartime activities.

This second organization had come to be as an intelligence organization for a Union Army lieutenant general (Winfield Scott), and eventually was put under the State Department, though Scott suggested putting it permanently under the War Department.

The first chief of the US Secret Service (under the Treasury) wasn't sworn in until July 5, 1865. And while their purview was still counterfeiting, it's suspected that they investigated other crimes in the United States, and that some agent/mission overlap with the other "Secret Service" (the National Detective Bureau) did happen. However, the Secret Service of the Treasury Department wasn't actually established as an official division within the department until an Act of Congress in 1882.

Only by the late 1800s and early 1900s was the Secret Service even nearing the purpose we know it has today. (Ansley, 93-95)^^2

So, knowing this, what exactly was the security even like? Did presidents just wander around willy-nilly with staff and not fear assassination? Obviously, this was less the case during the Civil War, once the National Detective Bureau became a little more involved in protecting the president, but it could be argued that peacetime security was still very lax oftentimes.

Interestingly enough, that didn't mean that care wasn't taken in planning assassinations. On July 2nd, 1881, President Garfield was murdered a mere 4 months after inauguration. The murderer, Charles J. Guiteau, shadowed Garfield for weeks to make sure that he would not be robbed of responsibility (he wanted attention, he claimed), and that the plan would be executed successfully. He shot the President multiple times in the back, after following him and seeing him separate from a friend on his way home. (Mitchell, 468)^^3

According to Kaiser, Secret Service agents were assigned for the first time to a President in the case of McKinley (so, 1901), but were largely ineffective. They were positioned poorly (only 3 were even assigned to him, and only 1 was usually around at a time) in Exposition Hall where McKinley was receiving guests. The only other security detail assigned to the whole hall were local detectives and a small military unit (Kaiser, 549).

Helen Du Barry (Helen DuB, by some names), wrote in a letter her account of the assassination of Lincoln, which also gives us insight into the security of Lincoln. She wrote that there were two "watchmen" who always accompanied Lincoln (since the war began), and dressed in "citizens' dress" (DuB, 1865)^^5

Through these stories, one thing is very clear and typically considered to be true. By today's standards, security was very, very lax. However, the President was a busy man at all times, and usually accompanied by people seeking to speak to him on issues, and it does appear that there were some people assigned to protect him, ineffective as they were (using him in this paragraph in a general sense, not a specific President).

Lincoln, especially, we know had a security detail, small though it was. McKinley, too, had a security detail, also small and very inefficient. Garfield, less so, judging by the details of that assassination in particular. One other thing is common: the security details of any size are and have usually been regarded as fairly ineffective when assassination attempts take place near public places and when the President is approaching a crowd/gathering, and 11/13 assassination attempts have been in these cases, and the same number have used handguns (Kaiser, 549). Kaiser even argues on that same page (549) that Reagan in 1981 was no better protected from the initial attempt (though he was better protected from subsequent injury, meaning it's argued that the first shot had the same chances as on McKinley but the other 5 bullets had far less chance on also hitting, thanks to better security) in that situation than was McKinley nearly 80 years early (also debatable, though).

Hope that sheds some light. I wish I could answer about those specific instances and pictures, but this should give you a general idea of how "secure" the President was, how the policies on protecting Presidents changed over time, and how security today compares to the security of 100+ years ago!

Edit: /u/saturnfan posted two other short stories that also answer the general question of "how approachable were presidents", which reinforce many of the above points and provide other insights :).

^^1 The Attempted Assassination of President Jackson: A Letter by Richard Henry Wilde Edward L. Tucker The Georgia Historical Quarterly, Vol. 58, SUPPLEMENT (1974), pp. 193-199

^^2 The United States Secret Service. An Administrative History Norman Ansley The Journal of Criminal Law, Criminology, and Police Science, Vol. 47, No. 1 (May - Jun., 1956), pp. 93-109

^^3 The Man Who Murdered Garfield Stewart Mitchell Proceedings of the Massachusetts Historical Society, Third Series, Vol. 67, (Oct., 1941 - May, 1944), pp. 452-489

^^4 Presidential Assassinations and Assaults: Characteristics and Impact on Protective Procedures Frederick M. Kaiser Presidential Studies Quarterly, Vol. 11, No. 4, Presidential Management: The Importance of Presidential Skills (Fall, 1981), pp. 545-558

^^5 Eyewitness Account of Lincoln's Assassination Helen Dub Journal of the Illinois State Historical Society (1908-1984), Vol. 39, No. 3 (Sep., 1946), pp. 366-370

While not nearly an authoritative answer, I can say pretty confidently that security details in the 19th century were not remotely close to what they are today. It was also far easier to see the President than it is today.

Take Polk for example, according to his diary, September 3, 1846 was allegedly the only day somebody didn't come to his office seeking federal patronage.[1] These people were regular civilians or minor politicians looking for federal appointments and went to solicit the President personally. To put it into perspective, it would be like a freeholder traveling to Washington to personally ask Obama for an appointment to some federal post. Even though Freeholders are elected officials, I doubt he or she would even get within a mile of the President, let alone speak to him in person.

In an amusing story, when Andrew Jackson was elected president, there was a huge party in the white house that resulted in thousands of dollars of property damage because random people from off the streets were flooding the executive mansion to party. If anyone really wanted to Kill Andrew Jackson, that was the day.

[1] Paul Bergeron, The Presidency of James K. Polk, 139.

Up until Lincoln there was almost no security for US Presidents. John Quincy Adams was known for his solitary walks and naked swims, which he did regularly. Andrew Jackson had been involved in a fist fight when a disgruntled discharged Navy officer charged and attacked him and then fled from the scene. Jackson was also the first president to have someone attempt to assassinate him. A White House painter tried to shoot him with two revolvers, but both misfired.

It wasn’t until the Lincoln administration and the Civil War when security started to be implemented. During Lincoln’s 1st inaugural address, soldiers were positioned in key strategic points around Washington, and police in plain clothes mingled in the crowds. Lincoln was very adamant about not wanting any security detail and was reluctant when he was forced to have guards. It wasn’t until 1864 that he finally accepted a security detail of four police officers to be his bodyguards. When Lincoln was assassinated he only had one police officer guarding his box, whom shortly after the play started, left his position and went to drink at a nearby saloon.

When Garfield got into office in 1881 not much had changed. There was still a security detail at the White House, but Garfield frequently enjoyed walks on his own. Even weirder at this time is that the White House was pretty much an open house. From the hours of 10:30 to 1:30, Monday-Friday, the White House was open for the public to come and talk to the president. Garfield’s future assassin often came to the White House to ask for a job. On the night before Garfield was assassinated, him and his Secretary of State, James Blaine, walked unaccompanied of any guards outside the White House. Garfield’s murderer, Charles Guiteau, stalked them, but lost his nerve to pull the trigger. The next day, Garfield and Blaine walked unaccompanied again, this time through a train station. Guiteau walked directly behind the President and put two bullets into him. The police weren’t even the ones who caught Guiteau trying to escape; it was a ticket agent working at the station.

Security got better after that but it was still very poor. In Grover Cleveland administration the total of police officers at the White House went from 3 to 27. Also, the Secret Service, which originally was created in 1865 to stop counterfeiting and other fraud, had informally extended their duties to partial protection of the President. When McKinley was assassinated in 1901 he did have a security detail, though they obviously failed him. After the third assassination in a time span of 36 years, Congress finally passed legislation to make the Secret Service the official agency for protecting the President. Over the next several decades, the responsibilities of the Secret Service expanded and their budget got larger. So to answer your question, during the 1800s the security detail for Presidents ranged from non-existent to placing very little minimal emphasis on protecting the President. It took three deaths for them to finally provide adequate protection.

If I'm not mistaken, that picture of Lincoln is during his review of the troops at Harrison's Landing, after the Battle of Malvern Hill. It appears from the following source that Lincoln was likely protected by McClellan's personal bodyguard unit, which is said to have always ridden gray horses. From "Notes of a Staff Officer of our First New Jersey Brigade on the Seven Days Battle on the Peninsula in 1862", source:

A few days after our arrival at the Camp, President Lincoln came down and reviewed the army. I presume by reason of the small space in which it was necessary to hold it each brigade was drawn up on the northern side of its own camp in double columns, closed en masse, and the field officers were dismounted... President Lincoln rode a large bay horse and was dressed in a black frock coat and a high silk had and rode at the head of the cavalcade with General McClellan and his staff of probably a hundred officers immediately behind him. They passed down from east to west along the front of the army, the President taking off his hat as he passed the colors of each brigade. (Pages 26 and 27)

Certainly members of that cavalcade would include the bodyguard the same author references earlier in an odd episode I can't find adequate explanation for:

There was an immense park of our wagons not very far from the hill the night before the Battle of Malvern Hill, and while the brigade was on the hill in line of battle and sleeping behind the breast works... Huggs says that about eleven o'clock while the General was walking up and down between the tree and the fire, the orderly on duty came up to the General and said that a messenger from General McClellan's headquarters wanted to see him outside of the rifle pit, and Huggs says that the General walked straight down that way, he, of course, not going with him.. He[Taylor]... had not lain down all night.

The morning after the battle of Malvern Hill... we went straight down to the James River and up along the river bank until we came to Berkley Manson, which was General McClellan's headquarters. We had an orderly with us and both dismounted and left our horses with the orderly... In about an hour an orderly called me. The General was standing on the porch, mounting our horses we rode off towards camp, I riding, of course, a horse's length behind the General. After going about two or three hundred yards, he checked his horse and said: "Ride up along side of me." Which I did. He then said: "Did you notice that I did not have my sword when I went to General McClellan's headquarters?" I said: "I did sir, I noticed that you had neither sword nor belt." He said: "You see I have got them now." I said: "I do, sir." He said: "Well, I got them at General McClellan's headquarters." He said: "Last night while you were asleep an orderly told me that a messenger from General McClellan wanted to see me outside the rifle pit, I went there and two men on gray horses met me, one of whom was dismounted. This man presented a pistol to my head and instantly demanded my sword. Believing that I was captured and a prisoner there was nothing else for me to do but give him my sword which i did. Upon taking it he immediately mounted his horse and rode off."

That is all that General Taylor ever told me on the subject, and it is all I know about. (I may add that General McClellan's body guard always rode gray horses). The fact is that this occurred, on my word as a gentleman and a soldier, exactly as I have stated it." (Pages 23 and 24)

At the time of publication the author was a superintendent of his state's veterans' home, and had been the head of the NJ Grand Army of the Republic, and a gubernatorial candidate. I can't find any motivation for the man to lie about something that seems so trivial, so why did General McClellan have his body guard disarm a brigade commander at gunpoint before the Battle of Malvern Hill?

Of context to Lincoln's later visit, if the Union Army's security services were apparently that paranoid in general, it's likely they were on hyper alert status while hosting the president in enemy territory.