In my opinion, I see it as a great duty that a civilian serve their country. Therefore I would be baffled by why so many U.S citizens protested the draft.
Short-answer: The draft granted exemptions to those with physical injuries and to those who were attending college.
Therefore, those who were drafted we're often lower-class/working-class teenage men who were living in a period of heightened racial tension.
It was a rich man's fight, but a poor man's war. This can by synthesized in an earlier time period actually. If you look further back into America's history, the Civil War's draft system for both the Union and the Confederacy were nearly the same.
If you were drafted in the Union, you either had to serve or pay a $300. Not many people could afford to pay that.
If you were drafted in the Confederacy, you had to have a certain amount of slaves to avoid being drafted. I'm going to tell you now that the South's hierarchical system had plenty of yeoman farmers without slaves.
In all three cases, it was the elites calling for war and the socioeconomically disadvantaged paying the price in blood.
I can further explain if you wish.
Source: Enduring Vision - Chapter 15 - The Crucible of Freedom Chapter 28 - Liberalism, Civil Rights, and the War in Vietnam
Why was there no similar opposition to the draft during WWII? Or was there opposition that has since been glossed over?
Edit: changed "backlash" to "opposition."