Most people assume that because there were Vietnam War protests, they must have happened immediately and for the whole war, which is inaccurate. As stated above, public support was strong for the Vietnam War from 1964 - 1968. During this time, people answered the draft call and some actually volunteered, as their Uncles and Fathers and Brothers had done for earlier wars. People believed the communists were an actual threat and that if they didn't fight in Vietnam, they'd have to fight closer to home later.
The Tet Offensive of 1968 changed all of that. We had been told the war was almost over, that "the light was at the end of the tunnel", when in fact, the Vietcong were planning a major attack in early 1968. Once that attack happened in January and February, Walter Cronkite, the most famous newscaster in America, declared the war unwinnable. A majority of Americans then turned against the war, and we saw a great increase in protests and anti-war sentiment, and draft resistance.
I think that as the war escalated in the decade, public dissent followed accordingly. I think we can start with the Johnson presidency, for while Kennedy did have troops in the region, the American public did not entirely perceive it as a war situation. When Johnson laid the case for war to the American people, for troop escalation, the public expressed its overall support for the cause. It was when the war's aims became more complex and more convoluted and when the casualty count began to rise in unexpected swelling that public outrage began to be louder and more deafening across multiple demographics. When the perception of deception from the White House resulted in a credibility gap, the reaction to the war became outright negative. This culminated in Johnson's democratic opponents gaining momentum in the primaries of 1968, and with the President not seeking another term. It was at this point when the public was fairly solid in its opposition to the Vietnam War.
I am assuming that you are talking about people in the United States and their reaction to the Vietnam War.
At the beginning of the 1960s, there was little US involvement in the war in Vietnam and it was not a big deal in US public opinion. Once the US got much more involved (especially after 1964) there was broad public support for the war. So the people reacted to it favorably. It was only later in the decade that the American people started to get tired of the war. They felt that it was being run poorly by a government that was lying to them. At this point, more Americans came to oppose the war and to protest it quite vigorously.
However, it is important to note that many, and perhaps most, Americans still supported the war.