That was probably the intent. AFAIK Blacklights usually use UV-A lighting though. It should be said, that UVA-A penetrates much deeper into the skin. You won't get sunburn from it, but it still can be damaging to cells.

It was previously thought UV-A would not be damaging, as the skin doesn't seem to be as photosensitive to it compared to UV-B or UV-C radiation, which results in red or brown skin. But newer studies suggest that the danger was underestimated. The cell damage just isn't as visible.

I don't believe the usual blacklight you find in a party cellar has the intensity to cause any damage though, but you may want to ask a dermatologist. You really need to be careful with dosage of light in all these wavelengths.