After all, if Jane Fonda can go visit the enemy during wartime (as shown here) without prosecution, there's no way they'd try to bring treason charges against protestors.

Treason is rigorously defined in the Constitution because the criminal charge was abused so badly by the British. The Constitution sets a very high bar to prove treason. That's why very few people have been convicted of it in over 200 years.

If someone merely protests military action, then they're acting fully in accordance to their 1st Amendment rights. On the other hand, if they actively try to hamper military operations, say by damaging railroad cars carrying military equipment or other such acts, then it could (but wouldn't) be argued that they're "adhering to said Enemies", in effect by acting as guerilla fighters on behalf of the enemy. In effect, they would be acting as agents for the enemy.