Not sure about AF, but at Ft. Lewis the Provost Marshall is in charge of that stuff. You have one of those? They have to provide on-post storage and registration of personal owned firearms. They are supposed to lock them up in a base storage facility and you have to sign it out and ask for permission to use it way in advance, on and on... Such a hassle a lot of folks that live on base don't do it. If you live on base, check with the Provost Marshall about storing, or ask your 1sgt, he should know. At Ft. Lewis, which is an infantry base where just about every building has an arms room, we stored our personal weapons in our own arms room and had it set up where we just signed a hand receipt on Friday afternoon.
Once the weapon has been registered with the Provost Marshall, and you sign it into the correct arms room and then sign it out properly, you can carry it anywhere you like basically (well, depending on your particular base rules) but it has to be unloaded and such. Bottom line is you CAN'T carry on a base. You can sign out personal weapons to use at the range on weekends or to take off base.
I know all this applied at Ft. Lewis, I used to bring my weapons on base everyday and never had a problem. Just make sure you know the post rules. There should be a copy in your arms room, go ask the arms room guy. He'll know more than some robot gate cop anyway, and aren't those civilians now? What do they know.
Contact arms room guy, 1sgt., and/or provost marshall office. Forget the police, the MP's, whatever, they don't do this the marshall does (in the army). But the army is very closely related to AF, so I'd be surprised if you couldn't keep 'em or carry 'em on post.
Finally, I think there is a federal law that requires military bases with firing ranges to make at least one of those ranges accessible to the public. In that light, you pretty much HAVE to have registration and storage.
Yeah, you need to contact some other folks. I'd be real surprised if that base was a "no firearms at all" installation. It is active duty, right? Whatever, find out what the equivalent of Provost Marshall is in the AF and contact that office, and go to your arms room and talk to them, they SHOULD have a list of all the base and maybe local and federal laws too. Ours had them laminated just outside the door.