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

GO to the Chaplain....The Chaplain can guide. Telling the Spouse, while emotionally satisfying, can cause that spouse to maybe commit an act of vengeance that can ruin all kind of lives especially if there are kids involved.
IF the Other Party is military, the Chaplain can notify the Chain and a No Contact can be issued and an investigation conducted.

Are you friend of the coworker or the one being cheated on? Is the troop deployed/deploying?

So my wife found out that my best friend's spouse was cheating on him while we were deployed. After a week of keeping it bottled up I told him he needed to check up on her as she has been acting funny as of late. He asked me to be direct and I said that we believe she was cheating on him but wasn't in the bedroom watching it ourselves. I asked him to be calm about it in case we were wrong.
In the end, she confessed, he went home early and attempted to work it out. They divorced a few years later but he thanked me since had it gone on any longer, the house would have been cleaned out had he redeployed at the end of the tour instead of part way.

I would have kept it to myself as unsubstantiated rumor but when I found out the dude was driving my buddy's car, that was the last straw.

Make the b**** famous.

Honestly I would not get involved unless asked to. Just better that way.

I think he should tell his battle. If he is a real battle buddy...tell him. I had this happen to a soldier one time. That soldier killed himself. I later found out that other soldiers knew. I asked them why they did not say something....they stated that it was none of their business. They didn't even care. This is not looking out for your battle. It is hard to tell but it needs to be told from someone. tell Top and let him decide...but do something.

Call her out publicly. Like at an FRG event. Whore

To be clear, Me and my co-worker are both Veterans. The spouse is civilian. We are concerned about the husbands morale and mission readiness. It would be a very bad situation if he is in the sand box when he finds out.

I say do nothing. After all, one our own Commander-In-Chief's did it, was caught in a lie about it, and still nothing happened to him.

I don't agree with what is going on, but at the very most, speak to the individual doing the cheating to tell them what they have observed.

Delicate situation. Silence and confrontation both have their pros and cons. For me, personally, I would have your coworker confront the spouse whom he thinks is cheating. Depending on what evidence he has, it may be an honest mistake or he may just be uncovering the beginnings of an affair that the couple could work through. Regardless, I'd argue that calmly telling the allegedly cheating spouse what he witnessed or heard could open a dialogue that would help clarify the situation.

If it turns out to be true, convincing the adulterous party to come clean to their spouse on their own often has the best results. If they don't want to come clean, then I'd consult with your chaplain, like SSG Ayscue mentioned. It's a tough spot to be in, for sure, but the chaplain would be able to help you; either directly, or by getting you in touch with people who can advise you more effectively than some random Specialist on the Internet. :P

Hope it all goes well for you, your coworker, and his friend. Let us know how it all goes down, if you can--I'm sure this isn't an uncommon situation, and everyone can probably benefit from your experience in this.

Wow, quite a slippery slope... As a divorced former Marine I can tell you that I would want to know. With that said I would also expect my confidant to sit me down with the following advice:
1. Make an appointment with a counselor or a chaplain (preferably both) right then in front of you. This is going to suck so you will need to talk to someone.
2. Research your state rules - THIS IS IMPORTANT! Each state is different regarding the division of property in divorce (which some consider your retirement). Additionally, some are more progressive with regards to who gets the children. As a dad who has physical and legal custody believe me when I tell you this! Many of us are stationed in a different state than our residence, keep that in mind. Also, how difficult will it be for her to attend court if it is in a different state? She may be forced to make concessions.
3. Do not say a thing until you have evidence. Hire a PI, the cost will be minute in relation to what alimony will cost.
4. File FIRST and in the state that BEST supports your position. If you reconcile - great...you can always pull the paperwork.
5. Immediately upon filing cancel/suspend ALL her accounts.
6. Don't pussyfoot around being nice. Establish your objectives and stand by them. Give nothing away, once a lawyer smells weakness they will exploit it. "No" can be used as a sentence. Use it often.

The big thing to keep in perspective here is to be certain what you want. I have seen women destroy my friends financial futures, and they were sad and pathetic like there was nothing they could do. If you want to save it...great. My advice is there are others out there and fortunately I recognized that and found someone much better!