What Rick says...
That's almost exactly how I do it. Vertical storage of screens under a cloth dust cover. Also, I normally run a dehumidifier at 50% when I'm not using the (also basement) darkroom. But I turn it off during film and print processing, letting the humidity rise to damp down the dust. For both film and prints this means off overnight so the materials dry slowly and evenly.
Rewashing should do the trick if the dust remained right on the surface. If not, I think it may be a problem, as the surfaces of photographic paper (my standard is MGIV air-dried glossy) can be much more delicate than I believe most people realize. Especially if one doesn't use a hardening fixer in order to make toning and washing more efficient.
Also agree with 'bdial' in that I dry prints face-up. And for the same reason as his.
Ken