What do you think the US government should do to those who illegally avoid the draft?

Put them in prison.

Explanation:

The mandatory enlistment in the armed forces of a nation is known as the Draft. The origin of military conscription is dated back to ancient Mesopotamia however, the modern draft first observed during the French Revolution in 1789. The United States of America also employed conscription during the Civil War which resulted in various bloody draft riots. The resistance to the draft reached its peak during the Vietnam War.

By the later phase of 1967, a total of 15,058 killed and 109,527 wounded in Vietnam.

Therefore, the US government should Put those in prison who illegally avoid the draft.