Since the manufacturers usually doesn't say (or if they do, they often outright lies), whenever I see purple light nearby, I prefer wearing UV-protective glasses just to be sure.
That probably is excessive. The UV from a Wood's glass "black light" tube is almost entirely in the UVA range.
It's probably not a good idea to stare directly at the tubes too long, but otherwise it's nothing to worry about if you are just passing through. If you were going to be around the lighting for more than thirty minutes, some eye protection might not be a bad idea.
How a Wood's glass fluorescent UV lamp ("black light") works is the shorter wavelength UV which is produced by the electric discharge in the mercury vapour excites the phosphor on the interior surface of the tube, which produces a longer wavelength of UV. The Wood's glass filters out most of the visible light, with the exception of a very small amount of violet (up to about 425nm) that makes its way through (mainly this is the mercury line at 404.7nm). The Wood's glass also filters almost all UV below about 360nm. So the UV being given off is almost entirely UVA. Most of the UV from the phosphor is a wavelength centered on about 370nm (mainly around 360 to 380nm). That is closer to what they would call "near visible UV".
UV tubes for skin tanning use a different type of phosphor with a shorter wavelength emission (330 to 370nm). In that case, eye protection is more important. (even though it still falls into the category of "UVA").
My LD32 UVC is novelty item, but when I use it I wear glasses, gloves and a jacket. I honestly don't know how dangerous the UVC from the LD32 actually is, but I take no chances and I respect it.
I would definitely consider some eye protection with those shorter UVC wavelengths. To be fair, the vast percentage of the UVC will be absorbed by the cornea of the eye it will not make it to the retina. But a small percentage could make it through, and this wavelength range is very damaging.
It should also be considered that at lower (visible) lighting levels, the iris of the eye opens more and lets in more light. This can make these lights more hazardous than sunlight, even if the UVC is at the exact same level of intensity.