How do I keep mold out of closets?

Damp Rid is fantastic for this. Get the kind that you hang in the closet. Keep an eye on them and swap out as the crystals "disappear".

I think most likely you have to much humidity. 35% to 40% is the target. You likely need a DE humidifier. I have used one in my basement for years, and use the water from it to water plants, and feed a HUMIDIFIER upstairs as we use a wood stove that dries the air.

Your bath exhaust fan should be run during showers and baths, and for a minimum of 20-30 minutes afterwards.

Towels, etc. should never be laundered with fabric softener, because it decreases absorption of linens.

If you have a bathroom window, and exterior humidity levels are low, open the bath window.

Run a dehumidifier, as suggested.

Dehumidifier is what you need. My friends in humid climates have them. Run the bathroom exhaust fan during and after showering. Open a window, even a portable fan will work.

Purchase a damp chaser and keep it plugged in during the Spring thru Fall. You can find them at hardware stores such as Ace. I keep one in my walk-in closet and bedroom.

Many good suggestions for lowering the humidity, but you must also leave the closet door open. The humidity is getting trapped in the closet which has no air circulation, and this is causing the mildew and mold. (PS: is your closet perhaps overstuffed so that air can't circulate around clothing?)