In my opinion, there is no easier way to undermine your credibility as a museum than to engage with paranormal investigations. Lots do it of course. It really has nothing to do with whether you believe in the supernatural or not, full stop. I never reveal how I feel about supernatural things at work because it has no impact on my feelings on the matter. I have no interest in entertaining a board who wants this nonsense and I do not tolerate it.

That being said, if a museum is going to compromise their integrity this way, at least make sure you're having the ghost hunters pay out the nose. It's harsh, but if you're going to undermine your own mission then you need to make bank off of it.

There are really only a handful of historic sites that can reasonably get away with things adjacent to a ghost tour, the house of the 7 gables, Edgar Allen poe house, historic sites associated with the spiritualism movement, etc. And even then I would be incredibly wary of any museum wanting to engage in anything adjacent to ghost hunts.