We spent New Years at NAS Key West. Their RV campgrounds are available for active and retired military, and federal employees. There are actually two camping areas: Sigsbee Annex is located miles from downtown Key West (or much of anything other than base housing). It containes 92 full hook up sites at $23 a night, plus a large number of dry camping "overflow" sites for $13 a night. When the hook up sites are full the hosts initiate a rotation system so no one is left dry camping for more than a couple of weeks. The other campground is at Trumbo Point, located just a few blocks from downtown Key West and Mallory Square. All of the limited number of spots at Trumbo are dry camping, $13/night.
Since hook up spots were all full when we arrived, we elected to stay at Trumbo since it is much closer to downtown and the tourist areas. The campground is a well marked grass field with reasonable spacious and level sites, very few trees, clean bathrooms and showers, even a pool ($1 entry fee). The main advantage of Trumbo Point is that the back gate in the Coast Guard section of the base is just blocks from downtown Key West. Just make sure that when you exit you ask the guard when they will lock it up for the night so you don't get caught outside and have to make the much longer hike around to the front gate.
We have a more detailed write up about our experience at Trumbo Point at our travel blog, feel free to read and comment there. http://shellonwheels.net/2015/01/07/and-we-begin/