I think by the late 1960s the question of why we were in Vietnam was a moot point. We were there. People began asking "when" and "how long"--When were we going to get out and how much longer were we going to allow our young men to be drafted into that war?
The Viet Nam War Era was one of social discontent on all levels.
Not only were folks beginning to question the War, but women and African-Americans were demanding their rights, as well.
People from all walks of Life were demanding answers to previously unanswered questions...How long will we really be in Nam?...Why AREN'T women paid the same as men for the same work?...How exactly DOES skin color make a person less than someone else?...
The Viet Nam War was the focus of very vocal and actvie protest because it was in our faces every night on the news. Those who experienced it will never forget the image of the little Viet Namese girl running down the road with her back on fire from the napalm.
And we will never forget the images of the Civil Rights marchers being attacked with fire hoses and police dogs as they peacefully walked down the road in Birmingham.
Viet Nam became not only the symbol for the "old way of doing things", it became the poster child for cross-cultural, cross-class questioning of our government's decisions and policies. It became the last straw on a very fed-up camel's back.
The 60s produced one of the most radical social revolutions in history. Changes in economy, technology, media, work roles, birth control, and many other things, produced dramatic shifts in society, particularly for young people. The 60s empowered many groups who had never had a voice before and young people took advantage of this strange new power. The 60s revolution challenged the concept of automatically respecting your 'elders and betters'. For the first time, television directly showed people the world and all its troubles (particularly Vietnam). And young people felt empowered enough to challenge it. 'The Draft', which sent young people to Vietnam even if they didn't want to go was a big problem for young people.
With their new voice, young people could speak to each other about their concerns, particularly through music. Bob Dylan (and others) radicalised a generation with a series of powerful, angry, anti-war songs. Youth broke away from conventional society and formed their own youth culture (hippies Woodstock etc). They started protesting the war. They convinced many older people that the war was wrong (which it was). The anti-Vietnam protests at home became a serious problem for the administration who found it very hard to present the war as worthwhile.
The reason Vietnam became so unpopular was because it was the hated symbol of 'the old way of doing things' during the powerful 60s social revolution.