Exactly how politcally active can an enlisted member of the military be?

Military members can vote, donate money to candidates or parties, and attend political functions/rallies out of uniform. They cannot work as a member of anyone's election committee, they cannot do any type of work for a political party, and they cannot speak about political issues in public while in uniform. They also cannot speak about political issues involving the military (e.g., Don't ask, don't tell; women in combat, etc.). to the public in our out of uniform.

The same rules apply for officers, but even off-hand comments against the President or members of Congress may be punishable by court-martial under the offense of contempt toward elected officials, which only applies to commissioned officers.

Same rules apply to officers and enlisted.

For Active Duty, you can't run for office. You can't appear at political rallies in uniform unless it is part of your duty. You can't use your military status where it could be seen as an endorsement. For example, don't write a letter to a paper endorsing a candidate and sign it with your military rank and title. You can donate to parties and candidates. You can volunteer in your off-duty time, but you need to be in civilian clothes.

If you are Reserve or Guard, most of the same restrictions apply. However, you can run for political office. Just don't campaign while in uniform.

never ever in uniform under any circumstances.

they cannot run for office while AD

same for O and E

They can be as active as they want, as long as it does not interfere with their military duties, and they do not use their being in the military as part of it.

And that goes for everyone in the military.