Having an exhaust fan is only part of the solution to mildew in your shower. Be sure the fan has been properly installed (exhausts to the exterior) and is large enough to do the job.

Mildew is an organism that flourishes in a damp, warm, and dark environment. If your shower stall is dark consider installing a recess light fixture approved for damp areas in the ceiling above the shower. The light and heat generated by the bulb should help.

Thoroughly clean up any mildew with a solution consisting of 1/3 cup of laundry detergent, one quart of liquid chlorine bleach, and three quarts of warm water. Allow the solution to sit until all of the dark stains have turned white. Flush the area with fresh water and towel it dry. After the process check to see if any mildew remains below the caulking. If so, the caulking should be removed, the area should be cleaned with the bleach solution and new caulking should be installed. CAUTION: Even though the solution is mild, wear safety goggles, rubber gloves and be sure plenty of ventilation is available. Never mix bleach with ammonia. The combination can be lethal.

As an added measure of protection, towel dry the shower after each use.