I vaguely remember a group of chasers (likely the ones who appear in the video) openly posting about seeking shelter on the base during the landfall last fall, so I'm surprised it's an issue now, if nothing came of it then.
That being said, I think their biggest transgression is posting footage of entering the base via a gate that's clearly unmanned. Without knowing all the circumstances, I don't necessarily fault them much for entering the base to find shelter. But to post that footage publicly is to advertise to the world that security was nonexistent at the base. This was either: 1) a conscious decision on the part of military personnel, who likely assumed no one would be insane enough to put themselves there during those conditions; or 2) a result of someone responsible for security abandoning their post (very unlikely, IMO). Either way, given how serious security is for any military facility, advertising this situation understandably creates big problems.
If the chasers had just done the bare minimum to find adequate shelter (perhaps filming for their own records) and then posted snippets that didn't make the situation on the base so obvious, I doubt it would be a newsworthy issue. Perhaps nothing will come of it anyway, but in my view, after seeing the video, they did just about everything they could to make themselves a target for scrutiny by people whose attention you'd rather not attract. Probably another good reminder for everyone to think seriously in advance about the consequences of sharing footage online that depicts anything sensitive or illegal... it's usually a needless risk if it doesn't directly involve extreme weather.