Sure, but there are insulation solutions that mostly prevent it. For example insulation foam layered between vapor permeable foil on the cold side and vapor inhibiting foil on the warm side.

But drywall (and other porous surfaces) are supposed to help regulate moisture, so if they never get an opportunity to dry because it's permanently cold and humid, yeah, that'll be a problem. Cracking a window at least 30 minutes per day (ideally on dry days) does wonders, though it won't help temperatures in winter.