The short answer is that the US supported South Vietnam because it saw it as a democratic ally in Southeast Asia and was concerned about the spread of Communist aggression, especially that from North Vietnam under Ho Chi Minh who wanted to reunite the country by force. Eventually, that escalated into support with military "advisors", troops, and war material as it became a full-blown proxy war.
For the most part, early on, people supported the US' position on helping the democratic regime of South Vietnam in principle. Yet when the war escalated into larger military operations such as bombings conducted against North Vietnam, people started to wonder if we were pushing our 'defensive' role a bit too far. This was turning into a 'real' war and there were questions on why we were becoming more and more involved later leading to widespread protests turning public sentiment against it.
Even after the events of the Vietnam War, there was a lot of negativity attached to it as the war that America "lost" as well as anything else associated -- which included many of the soldiers that came home.
I hope that this is the answer that you were looking for and it has helped you.
