There were two main ways in which the draft was controversial during the Vietnam War era.
First, the draft was controversial because there were a large number of people who did not want to participate in the war. As the US became more involved in Vietnam, the war became less popular. There were many people who felt that it was an unjust war. They did not think that they should have to serve in a war that they did not believe in.
Second, the draft was controversial because it did not take all classes of people equally. During the Vietnam Era, people who were well educated or who had money and connections were much less likely to end up getting drafted than other people. There were draft deferments for college students and even for graduate students, for example. Wealthier people could afford the sorts of legal advice that could allow them to find ways to evade the draft. The truly powerful could pull strings and get their sons into things like National Guard units that were unlikely to be called up. For these reasons, poorer people ended up getting drafted more than others. Many people felt that this was wrong and it led to controversy about the draft.
The Vietnam War draft was controversial because people who did not support the war and had no say in formulating war policy were nevertheless being forced to fight. Draftees could be under 21 (18 at the youngest), and at that time, nobody under the age of 21 was allowed to vote. Thus many 18-year-olds protested that they should not be forced to fight in a war if they had no say in electing the leaders who perpetrated it. Many of these protesters said there would be no conflict in Vietnam if the people who actually had to do the fighting had a vote. This argument led to lowering the voting age to eighteen.
The draft, as almost all drafts do, fell more heavily on poorer people. Part of this was due to the college deferment, which allowed people in college or graduate school to avoid being drafted as long as they continued their education. As richer people tend more often to pursue higher education, this more often left the poor to fight. The rich, as always, also had connections that could lead to safer assignments.
The lottery, in which people were called up on the basis of their birthdays, was considered by some inherently unfair. If the lottery said all men who turned 18 on May 3 would be drafted first and all born on May 4 would be 200th in line to be drafted, how was that just, these people asked?
The upshot was that in a democracy people resist being forced to fight and die in wars that seem unnecessary and that are also widely believed to be mismanaged. This was the first war, as has been often said, that showed up every night on television news shows in people's living rooms. Many were shocked by images of carnage and by men coming home in body bags. The graphic images also led to people resisting the draft and either saying, no, I won't go or no, I won't send my brother, son, or husband there. In addition, prominent people like news anchor Walter Cronkite came out against the war, which led to more questioning of the draft.
It should be emphasized that many also supported the war. However, even a sizable minority protesting a war and a draft can have a destabilizing effect on social cohesion.