Hey there, I can't speak to all American student protests/media responses in the 1960's, but I can tell you a bit about one specific protest and the subsequent media response. I am currently writing a paper on the Indian occupation of Alcatraz for a senior research seminar. This is my first post here (long time lurker) so I hope this is up to standards.
Some background for those unfamiliar with this episode. From November 20, 1969 to June 11, 1971, Alcatraz Island (home of the infamous former federal penitentiary) was occupied by Native American protesters. To clarify--the prison had closed in 1964. Other than a few caretakers, the island was essentially abandoned government property. The SF Board of Supervisors and the Federal Government were in the midst of trying to decide what to do with the island when the protest started. The occupation was organized by the United Bay Area Council of Indian Affairs and students from local universities (including but not limited to San Francisco State, UC Berkeley, San Jose State, and UC Santa Cruz). The protesters wanted to draw attention to the federal government's mistreatment of Native peoples, and were asking for Alcatraz to become an Indian community and cultural center. This demand later widened to include the actual title to the island. Due to factions, drug and alcohol use, and a total disinterest in negotiating with the federal government, the occupiers did not achieve their goal. The island is now a National Park and the focus is on the federal penitentiary years.
In terms of the media response, it shifted over the course of the protest. For the first month and a half, the protest was largely student run and during this time the media and members of the Bay Area community were in support of it. San Francisco Chronicle reporter Tim Findley was friends with a number of the protest's leaders and wrote numerous articles that explained the Indian's demands and the reasoning behind their actions. The community helped send food, water, and other supplies to the protesters, even dodging a Coast Guard blockade that was temporarily established. The island has no water supply so this was vital to the success of the protest. In January 1970, however, many of the students left the island to return to school and older, non-student Indians took over the occupation. This group of protesters changed their demands to include the deed, refused any form of negotiation or compromise, and eventually became antagonistic and militaristic to basically anyone not living on Alcatraz. As this shift happened, the local papers frequently described the unrest among the occupiers and ongoing feud with the federal government, but maintained a mostly positive standpoint. The protest did begin to get less media coverage, however.
A disastrous fire on the night of June 1, 1970 signaled a turning point in the media/protester relationship. Though no one was hurt, the fire destroyed five buildings, including the lighthouse. The light was a popular and comforting sight on the Bay's horizon and an essential safety feature in the dense SF Bay fogs. The Chronicle's editor and several reporters contributed generators and some manpower to get the lighthouse operating again, but that was the last big contribution from the local community. As the occupation became more militaristic, they became increasingly hostile towards reporters and camera men who attempted to visit Alcatraz. The cause of the fire was determined to be arson, but the perpetrators were never found. The government, naturally, blamed the protesters and the protesters blamed undercover government agents. The Chronicle gave more coverage to the government's story and decried the protesters for refusing to allow Coast Guard fire crews onto the island.
In December 1970, the Chronicle's editor agreed to a government request to favor the feds. A White House memorandum quotes him as such: "the whole Alcatraz occupation has soured... the present Indians [are] a filthy, drunken, drugged lot." Additionally, they refused to return his generators. Public safety was also becoming a real concern. The Indians had threatened the Coast Guard and work crews with projectiles, shot at Coast Guard boats, and actually fired arrows at a tour boat.
All this being said, many of the protesters and organizers feel that the media always unfairly maligned them. I don't share this viewpoint; I think they ruined what started as positive media coverage and attention through their hostility. If you'd like to learn more about this, I suggest Alcatraz! Alcatraz! The Indian Occupation of 1969-1971 by Adam Fortunate Eagle or American Indian Activism: Alcatraz to the Longest Walk edited by Troy Johnson, Joane Nagel, and Duane Champagne.
For great coverage of the whole occupation, read Troy Johnson's The Occupation of Alcatraz Island: Indian Self-Determination and the Rise of Indian Activism.
If you want to get a snippet of how the newspapers were covering it, check out the San Francisco Chronicle. Fair warning: I've not found a digital archive of the Chronicle that goes back further than the mid 1980's. I found all of these articles on my University's microfilm collection.
TL/DR: The media and public initially supported the student led phase of the Indian occupation of Alcatraz, but hostility on the part of the protesters towards outside forces made them unpopular. Additionally, a fire on the island destroyed a beloved landmark and the drug use among the Island's occupants further turned the public off to their cause. When the government requested that they be given more favorable coverage, the San Francisco Chronicle agreed.
I want to write a little about the "general population response" with a specific example found in the midwest: The protests in the late 1960s in Madison, Wisconsin.
Madison provides a nice example because it was a mix of middle-class, working Americans with high school degrees, as well as young, idealistic college students. The police force at the time was conservative, and didn't sympathize with the generally more liberal students.
In late October of 1967, the Dow Chemical company (maker of "liquid fire" napalm) came to Madison to do jobs interviews on campus. A group of students showed up to interfere and protest due to napalm's use in the Vietnam war. Campus leaders, inexperienced with handling unrest, were caught between the wishes of conservative financial backers and liberal student protesters. They called in the police, who were also untrained to deal with riots. The result was a chaotic fight, and the police used billy clubs to beat students both male and female. Not all students were liberal, or sympathized with protesters, and these conservatives watched and cheered police as they swung their clubs.
Among the students protesting that day was Paul Soglin, who emerged as a leader during the anti-war protests. Six years later, at the age of 28, he was elected as the mayor of Madison.
Many students were radicalised after seeing violence used on fellow students. Participation in anti-war protests grew. This included on-again off-again student Karl Armstrong who went on to joined 3 other students in planning to bomb the Army Mathematics Research Center (AMRC). The AMRC occupied a few floors of a campus building, Sterling Hall. They didn't intend to kill anyone, so they planned to set off the bomb in the middle of the night. At about 3am, on Aug 24 1970, they drove up to the building with a vehicle filled with nearly 2,000 pounds ammonium nitrate and fuel oil. The resulting blast damaged a few dozen nearby buildings (mostly windows) and completely destroyed several floors for Sterling Hall. It also killed 33 year old post-doc Robert Fassnacht, who was there working on research into superconductivity.
More info: The Sterling Hall Bombing.
Books:
In The Movement and the Sixties: Protest in America from Greensboro to Wounded Knee, Terry H. Anderson relates a more generalized narrative rather than utilizing a specific event/group/leader as a microcosm of the protest movements in that era. I read it in college and your question prompted me to pick it up again. Here are a few quotes from the Preface that pertain to your question. I'm going to continue reading it to see if I can provide a more comprehensive answer in the near future.
On academia's response:
Intellectuals also have feuded over the sixties. During those impassioned days, Seymour Martin Lipset, Nathan Glazer, Daniel Bell, Irving Howe, Sidney Hook, and others characterized student protesters as irrational, irresponsible, anti-intellectual extremists. William O'Neill wrote a critical history of the era, Coming Apart, and historian Richard Hofstadter labeled the era "The Age of Rubbish." Other academics were more positive. Historian Howard Zinn spent time with civil rights volunteers in the South and wrote of their "unquenchable spirit." Sociologist Helen Swick Perry experienced the counterculture in San Francisco and was attracted to hippie philosophy, while Theodore Roszak added that the "young, miserably educated as they are, bring with them almost nothing but healthy instincts."
And some comments on the media:
Attending undergraduate and graduate school in the late 1960s and most of the 1970s, I read a few popular histories on the sixties, and realized then that the authors had misunderstood and distorted the movement. These commentators were older, and perhaps their interpretation was a natural result of the passion of the decade, but also it was a consequence of the sources they employed. Invariably, they relied on the mainstream press, which would be appropriate for many topics, but not for all forms of social activism in the sixties. The movement was a revolt against the establishment, and major newpapers were a pillar of the foundation. Editors often published slanted reports, and that was especially true of articles about antiwar demonstrators and campus activists at mid-decade and later about the counterculture, black power, and women's liberation. During the Berkeley Free Speech Movement, for example, the San Francisco Examiner and Oakland Tribune wrote that the student activists were communist-inspired fanatics. Later, when 800 uniformed servicemen and activists protested the war by marching on the streets outside of Fort Hood, Texas, the Austin American-Statesmen reported that patriotic local residents confronted demonstrators and prevent them from completing their parade, which activists noted was not true, another example of "media lies." Because activists did not trust the press, they formed their own, establishing over 600 underground newspapers which by 1970 had a circulation of about five million: and they established at least three national wire services while printing up libraries of posters, leaflets, and newsletters. Generally, their reporting was fine on issues they supported, such as the civil rights struggle, and later in the decade their articles became more critical toward the government and more tolerant toward activism. These establishment sources have been used. Yet one cannot neglect the underground sources, as almost all authors have done so far, for that is tantamount to writing a history of the American Revolution by examining only British documents--and not those of the rebels.
Todd Gitlin's book, The Whole World is Watching, speaks to many of these issues.