I find a good bathroom fan helps to cut down on moisture and mold problems. I've been installing this one by Broan lately.
http://www.lowes.com/pd_10064-14-QT140LE_1z0z4ih__?productId=3747685&Ns=p_product_price|1&pl=1&Ntt=broan
It uses a 6 inch duct instead of a standard 4 inch, so it sucks a lot more air. You can wire it directly to the light, that way if someone is using the bathroom they will never forget to turn on the fan. It's pretty quite and I have had no complaints when doing that. Or you can install a humidity sensing switch like this one.
http://www.amazon.com/Leviton-IPHS5-1LW-Humidity-Sensor-Control/dp/B00H3QQD64/ref=sr_1_1?ie=UTF8&qid=1435501004&sr=8-1&keywords=humidity+switch
That way the fan will come on when the humidity level is high and keep blowing until the humidity level is reduced. Even if the person has left the bathroom and turned out the lights, then fan will stay on until the humidity is gone.
There are some fans that have the humidity sensor right on the fan, but hey cost a lot more. I like the features on this one, but $377 is a bit too much for a cheap rental. Maybe on a high end unit it would be ok.
http://www.lowes.com/pd_539628-14-XB110HL_1z0z4ih__?productId=50061657&Ns=p_product_price|1&pl=1&Ntt=broan
I also have been using some mold killing primer in bathrooms. I don't know if it works or not, but I figure it's worth a try.
http://www.lowes.com/pd_617619-90-276049___?productId=50290469&pl=1&Ntt=mold+primer
Many older bathrooms will have windows in the showers in my area and I prefer to remove them when I get a chance. They are a maintenance headache. Water sits on the window sill and warps the wood. Replacing the wood sill with a piece of granite or stone helps. But I'm always caulking them, and they lead to water and mold damage behind the shower surround. I have 2 left and as soon as those tenants move those windows are coming out.
I use the caulkless tub surrounds. We love them. Easy to install, low maintenance. Menards has a great selection of them.