I am currently on active duty and have lived on base for most of my 20+ years of service (currently live on Hickam AFB).
Quick answer - it depends, but you are most likely fine if you are smart about it. Deep in on-base housing, away from the flight line or any obvious radio installations - you're probably fine. Bring a copy of your license if a MP or someone else asks what sort of radio you have in your car.
I'd be really, really surprised if the MPs or PIO had any clue in the world what you were talking about if you asked about ham radio. Thus, the safe answer you will probably get is "no" despite what the real answer might be.
In Guam (3 separate assignments), there was legacy regulation in place for Amateur Radio, but it took a while to get permission since no one had pursued it in years and those filling the approval positions had no idea what Ham Radio was. It took a while to get approval since it's easier (and often safer) to say no than to say yes.
Here at Hickam, all military housing has been privatized unlike in Guam. With privatization comes commercial CC&R covenants, AKA no antennas allowed whatsoever except satellite dishes. That said, I know a couple hams who have stealth antennas in housing and have had no issues.
73, Dave N2NL