Two years ago at Lackland AFB, me and my two buddies all had scanners, and we noticed quickly we were being shadowed by some security forces personnel for about 20 minutes when they finally came up and confronted us. Their only concern was if it could transmit, and the airman looked my scanner up and down turning it over and over looking for a PTT. As he was holding it, he asked me what I was listening too and I told him "the planes doing the demos," of course right at that exact moment, I see the LED color light up for the Security Forces system I was also monitoring(luckily the volume was down when he had it and he didn't read the alpha tag). We were left alone after that.
The next year the airshow was held at the nearby Randolph AFB, and they couldn't have cared less about the scanners. When we were wondering around, we found where the announcers booth was and could watch him talking to the pilots and hear what both were saying. A lady involved with the airshow organizers was the only person who asked about the scanners, and only to ask if they could transmit. She then asked if we were listening to their frequencies, and wanted to know what they were to confirm it. She said they had 2, and we rattled off the VHF one right away, and she started getting smug when we were giving her other VHF freqs in use that day, and said their other channel was UHF, which I popped off the correct channel to her right off after she said "UHF"(having found it about an hour or so early on a service search on the PSR-500) and it made her jaw drop, like we had discovered a secret frequency or something.
Seriously, having a scanner is the best thing to accompany you to an airshow because you always have the heads up about when the cool demos are happening and when the planes that are doing flybys only and what direction they're coming from, and also you get to hear exactly why and how some demos are delayed. Like it was said though by others, it really is up to the base commander to either ban scanners or not, even arbitrarily and without reason, and we just have to make due with their rules. Of course there's nothing they can do about you sitting in your car outside of the base listening and watching(albeit from a short distance depending on where you're at).