Is it better to leave the door open or run the fan while showering?
The fan is better, as it actively moves air out of the room.  However, if you're getting mold with the fan on, then it's either undersized or not working properly.  Keep in mind you may need to let the fan run for a time period after you actually finish the shower to get all the humidity out of the room.  You can get fans that speed up and down depending on the level of humidity, and even turn off when it is below a certain amount.