It's kinda hard to understand from a modern perspective as needing or using a draft is such an alien concept to us, but back in ye olden days it was seen as your duty to your country and something you accepted as a possibility by choosing to remain a US citizen.
There were plenty of people who opposed the war and didn't want to fight but still accepted being drafted as part of their responsibilities as a US citizen. "You might be drafted" was an established part of the law and the people who dodged were seen as basically saying "Nuh uh, the law doesn't apply to me!" and avoiding both the obligations that everyone else accepted and also the consequences of their choice by running away.
Also a lot of the 'Nam draft dodgers were part of the hippy movement and with that also outright communist sympathizers, which wasn't... great for public relations at the time.