I have a coworker with direct knowledge of a Military spouse cheating on her husband. What action, if any, should he take?

Unit Chaplain's are trained to deal with these subjects specifically Military Related seek his counsel

The Chaplains office is the best first step.

She should mind her own business unless she has pictures or other solid evidence or she is the one. Perception is not always reality and interfering in other peoples lives often does more harm than good. If she does have physical evidence then go to the spouse and tell them.

Depending on the situation of both parties. I would suggest that they both see their Chaplain or Pastor. If needed for counseling. Was this a one time offense or multiple times. How emotionally stable are the members. Consider family situations also are they looking for separation. Chaplains and Pastors are best suited to help but also you can be there for support.

Speak directly to that individual and remind them of the damage they are inflicting on their self, family and true friends by carrying on this relationship. Remind them that the action is damaging to their career because it paints them as dishonest. If there really is a problem in the marriage, and the affair is "the drug of choice", they need to sit with a counselor, the both of them, and hash things out and make the necessary changes or truly move on.

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Do what you think is right to end what is going on. Think out side the box.

I would definitely inform the military member. They have every right to know what's going on behind their back. As hard as it is to do, speak without judgement and sincerity to their emotions. If they dent it/don't want to believe it, you can at least have given them a warning about what to expect. Let's face it, cheaters always get caught and if they eventually get caught or end the relationship over the new fling, he has been warned.

It would hold more weight if there were two people with direct knowledge. Without a pregnancy or an eye witness it is very hard to prove.

IAW the Manual for courts martial, adultery must have 3 elements.
(1) That the accused wrongfully had sexual intercourse with a certain person; (2) That, at the time, the accused or the other person was married to someone else; and (3) That, under the circumstances, the conduct of the accused was to the prejudice of good order and discipline in the armed forces or was of a nature to bring discredit upon the armed forces.
It is number 3 that usually lets a command decide when not to punish. It is number 1 that is most difficult to prove.
Those in the Armed services saying not to do anything, you are liable to be punished for "Failure to Take Appropriate Action" In violation of AR 600-20, Para. 1-5 and 4-4; AR 600-100, Para. 2-1 and 2-23.