Many, perhaps most, people believe that young people were more likely to oppose the Vietnam war than older people.  This is not surprising if you judge from the visible protestors, many of whom were young.  But this is misleading, because the most visible and active people in any movements are likely to be young, since they have the time and freedom to engage in protests.  The most active supporters of Goldwater in 1964 were often young, as were the peace protestors just a few years later.

There were many polls on public opinion during the war, and they show a consistent pattern by age.   Young people were more likely to support the war at the beginning, when it was popular, and more likely to support it at the end, when it was not.  The polling organizations used many different questions to tap public opinion during the war, often changing them as events changed.  No matter what questions were used, the same pattern of public opinion was found, with the young people more likely to choose the hawkish alternative(s) than older people.

(There was, by the way, substantial support for a stronger war effort, especially early in the war.  For instance, in a poll conducted in February 1968, 25 per cent wanted to "gradually broaden and intensify our military operations", and 28 per cent wanted to "start an all-out crash effort in the hope of winning the war quickly even at the risk of China or Russia entering the war".  Just 24 per cent wanted to "discontinue the struggle and begin to pull out of Vietnam gradually in the near future", and 10 per cent wanted to "continue the war at the present level of military effort".  So, much of the disatisfaction about the war came, early on, from the belief that not enough was being done to win it.)

There is just one question that was asked, with the same wording, throughout the war.  Gallup asked the following question frequently: "In view of the developments since we entered the fighting in Vietnam, do you think the U. S. made a mistake sending troops to fight in Vietnam?"  If some one answers no, then we can assume that they supported the war.  Almost every time the question was asked, people under 30 were more likely to say no than people 30-49, who in turn were more likely to say no than people 50 and older.  The two exceptions were within sampling error.  (The numbers for those who agreed that the war had been a mistake are, essentially a mirror image of those who did not.  Those with no opinion started at about 20 per cent and declined as the war went on, though there were always more in the oldest group.)  Here's a table with the data:

Other common beliefs about public opinion on the Vietnam war are also false.  Educated people were more likely to support the war, not less.  There is not as much data on the subject, but draft status did not seem to affect opinions on the war.

Some common beliefs about the war are correct.  Women were more dovish than men, and blacks more dovish than whites.  All the patterns that I have mentioned were also found in public opinion during the Korean War and World War II.  The science of polling is young and so we do not really have good data from before World War II, but there is every reason to believe that, in nearly all wars, young men are more hawkish than older women.  We are much less warlike now than we were a century ago because we are older and women have far more political power.  (If you are interested in learning more about public opinion on the Vietnam war, I would recommend John E. Mueller's War, Presidents and Public Opinion.  It is out of print, but should be available in any university library.  Full disclosure: I once took a international relations course from the author.)