Besides using a fan to get the humid air out of the bathroom, I'd paint the area where the condensation drips down with a paint specifically meant for bathrooms.
Bathroom paints, like Zinsser's PermaWhite Bathroom Paint, will contain mildewcides. These mildewcides have an affinity for water, and under wet or humid conditions, they actually migrate out of the paint film to the surface of the paint where they kill any mildew before it has a chance to grow.
So, by using a bathroom paint on those damp surfaces, you can at least remove the symptoms (mildew) if you can't cure the problem (condensation).
But, keep in mind that the mildewcides will leach out of the paint and be dissolved in the water, so the paint won't remain mildew free for very long under those conditions. Try to get a bathroom paint that has the highest gloss level. Those will use a stronger mildewcide that leaches out more slowly, and that's what you would need of the surface of the paint is actually getting wet with running water.
Also, because of the way that mildewcides work, putting on multiple coats of the stuff will make the surface remain mildew free for longer. It's a case of diminishing returns, however, because the mildewcide has to migrate out fast enough to ensure there's enough at the surface to be effective at killing mildew spores. The further it has to travel, the less of it is gonna travel the full distance to get out of the paint.
But, that's what I'd do anyhow.
