What’s the law if you take a photo of a US base or personnel from a public place, not on the base?
Technically, it belongs to Uncle. Practically, unless they have stuff they didn't want you to see, like stealth fighters or UFO’s, it's usually not a problem. For example, if you go to base where they're having an air show, unless they tell you no cameras, it's ok. I've only seen a camera ban at an air show once, and that was because they had an F-117 on display, back when they were still very new.