Important point: I highly encourage everyone to read the response to my answer by u/XenophonTheAthenian. They make very valid criticisms of my comment and the assumptions about late Republican politics which I’m making. When I saw the question this morning I couldn’t resist offering a response, and while I defend my answer as an acceptable popular history, it doesn’t adequately engage with contemporary scholarship (partly because it was written on a phone, on a bus). I intend to thoroughly revise my answer and also justify some of my assertions which Xenophon has critiqued. This will take time so please bear with me. In the meantime thank-you for the appreciation of my answer and the fantastic discussion it’s generated.
I would argue that in general wealthy Romans of the late Republic took a very personal and direct interest in the lives of ordinary people. I’m going to discuss client-patron relationships briefly, but bear with me. I promise it’s relevant. I’ll also add that I’m going to be discussing relationships between male Roman citizens, albeit from very different classes. There is much more to be said about non citizens, foreigners , women and enslaved people. Finally after reading back my answer I think I’ve given a somewhat rosy picture of the early republic. Bear in mind the notion of the early Republic governed by self supporting farmers was an ideal and that bribery, corruption and patronage were always present, though I’d argue they reached much higher levels during the first century BCE.
At the time the tv series Rome is set (roughly 50-31 BCE, even though nobody ages 20 years), the Republic had gone through a number of crises which had shaken it to the core. The fall of Carthage and Corinth in 146 BCE had led to a decline in the power of independent small farmers as an influx of slave labour meant wealthy landowners could easily outcompete them. This class of large landowners could then use their wealth and power to buy up or force farmers off their land, expanding their holdings even further. This economic rebalancing led to large numbers of people migrating from the countryside to Rome and the rise of a super rich oligarchy.
Traditionally independent farmers had been the backbone of the Republic, supplying its armies and participating directly in politics. The historic ideal whom many looked to was the Dictator Cinninatus, who was supposedly ploughing his field when envoys from the senate asked him to return to office. The decline of the yeoman farmer meant that military forces had to be raised from the urban poor who would require some kind of compensation (usually loot or grants of land) for their services. Generals, who were drawn from the wealthy aristocracy, were thus incentivised to cultivate positive relationships with their troops as their loyalty depended more on reward than a higher notion of service to the Republic.
At the same time the same wealthy oligarchs dominated politics through similar tactics. Lower class Romans (Plebeians) were cultivated by the Equestrian and Senatorial classes and incentivised to vote against political programmes (such as land reform) which went against the aristocracy’s interests. In exchange they would receive financial and social assistance from their aristocratic patron. An influential Roman Senator might have hundreds of these clients on his books, many of whom would arrive at his home every day seeking largesse. A patron might also perform services like ransoming you if you were captured by pirates, or protect your family while you were away on business. In exchange for this help, clients would be expected to support their patron politically and provide practical assistance (whether that be helping with building work, or forming a mob).
What all this boils down to is that much of late Republican politics was based on the urban poor entering into client-patron relationships with the wealthy, whether this be supporting a particular general in the hopes of loot, or offering your vote in exchange for money. It also meant that politics was in some sense reduced to transactions between individuals which required the rich to interact with the poor and attend to their needs. As the Numidian rebel Jugurtha was alleged to have quipped ‘Rome’s a city for sale and bound to fall as soon as it finds a buyer’
People like Caesar, Pompey and Crassus knew that to succeed in politics you needed to take an interest in the personal lives of thousands of people and interact with them on a daily basis. In many cases this would simply be a bribe which you regularly paid someone, however as a patron it was also in your interest to take an interest in your client’s personal life. You might be the guest of honour at their wedding, speak on their behalf in court or take on their children in your service. I would speculate that many of these relationships evolved into socially unequal friendships or at least friendly acquaintanceships.
In the realm of the army Caesar in particular was keen to identify himself with the common soldier and spoke of his men as comrades, which linguistically at least put him on the same level as them. He was also eager to praise and reward good service as the real life case of Pullo and Vorenus demonstrates. The first two minutes of episode one of Rome are adapted from Caesar’s Gallic War Book V. Although it’s recounting an extraordinary event, Caesar’s account is notable for the interest he takes in the personal lives of two of his men.
In that legion there were two most gallant centurions, now not far from the first class of their rank,6 Titus Pullo and Lucius Vorenus. They had continual quarrels together which was to stand first, and every year they struggled in fierce rivalry for the chief posts. One of them, Pullo, when the fight was fiercest by the entrenchments, said: "Why hesitate, Vorenus? Or what chance of proving your pluck do you wait for? This day shall decide our quarrels." So saying, he stepped outside the entrenchments, and dashed upon the section of the enemy which seemed to be in closest array. Neither did Vorenus keep within the rampart, but in fear of what all men would think he followed hard. Then, at short range, Pullo sent his pike at the enemy, and pierced one man as he ran forward from the host. p293 When he was struck senseless the enemy sought to cover him with their shields, and discharged their spears in a volley at the foeman, giving him no chance of retirement. Pullo's shield was penetrated, and a dart was lodged in his belt. This accident threw his scabbard out of place, and delayed his right hand as he tried to draw his sword, and while he was in difficulty the enemy surrounded him. His enemy, Vorenus, ran up to him and helped him in his distress. Upon him at once all the host turned, and left Pullo, supposing him to be slain by the dart. Vorenus plied his sword at close quarters, and by slaying one man drove off the rest a little; while he pressed on too eagerly he fell down headlong into a dip in the ground. He was surrounded in his turn, but Pullo brought assistance; and both, unhurt, though they had slain several men, retired with the utmost glory within the entrenchments. In the eagerness of their rivalry fortune so handled the two that, for all their mutual hostility, the one helped and saved the other, and it was impossible to decide which should be considered the better man in valour.
edit: I originally wrote that wealthy oligarchs supported land reform. This was a typo, I meant to say the opposite. I originally wrote that Jugurtha was Nubian, this was also a typo: I meant to write Numidian. Thanks to u/neaaopri for pointing that one out.
There's no very good answer to this question, unfortunately. We're quite sure that the Roman nobility took stock of the so-called "lower class." Even without having to point to nomenclatores and so forth the current models of Roman politics emphasize the importance of interaction between and communication with the populus Romanus (however that was conceived, a controversy in its own right), and this recognition has underlain the current controversies in scholarship on the subject of Roman "political culture." The textual evidence for fairly close interaction between nobility and "ordinary" man is quite abundant, but the question is why exactly it mattered.
The old model, which /u/NumisAI relates, is rather pejoratively called these days the "Frozen Waste," based on a criticism by John North in 1990. This model, backed up by reams of prosopographic studies, was championed by Friedrich Münzer and Matthias Gelzer, two of the greatest German classicists ever to have lived, and is best known in English language scholarship from the work of Syme and Scullard. The Frozen Waste basically works thus: Roman politicians were blue-blooded aristocrats who formed alliances with each other, often by marriage but also in less formal ways. Subordinate to themselves they could muster hundreds of clients, obedient and willing to engage in whatever political actions were required of them, thanks to the strong bonds of reciprocity that underlay Roman society. As a result, Roman society was strictly oligarchic, and the Roman nobility paid attention to the Roman people not as populus Romanus but as clients.
The Frozen Waste is, to put it mildly, not accepted by anyone anymore. Fergus Millar wrote a very influential article in 1984 called "The Political Character of the Classical Roman Republic" in which he argued that there's very little evidence for client-dominated Roman politics and suggested that the contio, public speeches given in the forum before the populus Romanus, was the central element of Roman political culture (though this term was not in use until 1990 when Harris first used it). Millar has rightly been criticized for being too eager to compare the Roman Republic favorably with the Athenian democracy, but his work pretty much torpedoed any support for the Frozen Waste, which was already thawing in the mid-80s. By 2001 Mouritsen could say without any controversy that the emphasis on clientelae was dead.
The issue is that there isn't really a consensus that's emerged from this. Current scholarship for the last two decades or so has been focused on the question of how important communication between the Roman nobility and the Roman people was, and how exactly it functioned--a question that, under the Frozen Waste, would have been unthinkable. But there's no consensus here. The introduction of Cristina Rosillo-Lopez's new book Public Opinion and Politics in the Late Roman Republic, if you have access to it, does a good job of summing up the state of the field on this subject right now, and it's a bit beside the point so I'm not going to go into it. The point is that we know that there was a great deal of communication between the Roman nobility and the populus Romanus--again, however that was conceived--but that we don't know whether it was predominantly two-way, one-way, or what. Roman politicians spoke at contiones towards the people, but does that mean that the people spoke back? Mouritsen would emphasize that the people didn't have any agency of their own, whereas Morstein-Marx points out that the textual evidence repeatedly makes reference to the way that speakers at contiones needed to respond to and size up the attitudes of the crowd.
As for Caesar specifically, there's another controversy there. Simply put, we don't have any idea how Caesar was published. We're not even sure what the BG and BC are supposed to be, as texts. The Vorenus and Pullo passage gets a lot of attention, but nobody's quite sure what to do about it. People like Goldsworthy use it as an example of Caesar's interest in the individual soldiers, for whatever purpose. It's certain that Caesar makes note of a lot of individual names, as well as their actions. But there's not really any direct parallel to the Vorenus and Pullo passage. It's quite long, it comes as the climax to the siege of Q. Cicero's camp by the Nervii (an episode that Riggsby finds particularly important in Caesar's overall narrative), and in context it's a juxtaposition between Gallic and Roman military virtue--a point that, incidentally, HBO's writers totally misunderstood, mistaking the point of the passage as demonstrating the superiority of Roman discipline, which is exactly the opposite of what's going on. As is typical, Caesar gives essentially no further explanation of the characters, their names are simply mentioned and then they're discarded. We're not even sure if Pullo's name is Pullo, the manuscripts have like five different variations. Pullo is mentioned in the BG briefly, but Caesar doesn't even make the connection between the two passages. More typically the references to individual soldiers--who are almost always legion commanders, military tribunes, or centurions, and essentially never "ordinary" soldiers--are very brief, and as in the Vorenus and Pullo passage Caesar basically only tells us about their careers. Caesar, mind you, wasn't even present at the incident in question. When Q. Cicero's camp was under siege by the Nervii Caesar was miles away, and presumably the various descriptions of the siege, which is quite unusual in Caesar's narrative, were coming mostly from Q. Cicero. But the short version is that we're not really sure what the point of passages like this in Caesar is exactly.