This is a condensation problem, and there are several possible solutions. The first thing to do is simply eliminate the mildew. Mildew is a toxic organism that can be very dangerous to all who use the bathroom. Clean it with a one-to-one solution of water and regular laundry bleach. Be sure to wear gloves and goggles to protect yourself. After you have cleaned the wall and ceiling, and allowed them to dry properly, apply a paint that specifically contains fungicides. This will protect the surface against future mildew better than a regular paint. The mildew has formed because the bathroom is not properly ventilated. Next, an exhaust fan is absolutely essential. If the bathroom does not have a window, there is no other possible way to allow the moisture to escape. Check the insulation in the attic just above this area. Little or no insulation will cause the condensation and mildew to return. If you cannot get to the attic, add rigid foam insulation on the bathroom ceiling.