How to best keep floor vent by shower dry/mold free?
Cover the vent
They sell magnetic vent covers at nearly all hardware stores. Buy one, and cover the vent when it's not being used. This will prevent moist air from entering the vent, though it may not stop water if it pools around the vent.

Install an exhaust fan
Install a properly sized exhaust fan in the bathroom. Let it run during showers, and for about 20-30 minutes after showers.
WARNING: Make sure you vent the exhaust properly.
Sizing an exhaust fan
Exhaust fans are sized using Cubic Feet Per Minute (CFM), so you'll have to first figure out how many cubic feet your bathroom is. To do this, measure the length, width, and height of the room.


Length = 10 ft.
Width = 8 ft.
Height = 8 ft.


To find the cubic feet, simply multiply these numbers together.

10 ft. * 8 ft. * 8 ft. = 640 ft.3

Next we'll divide the cubic feet by 60, to figure out how many CFM is required to exchange the air one time in an hour.

640 ft.3 / 60 Minutes = 10.66 ft.3/Minute.

It's recommended that the air in a bathroom be exchanged 8 times per hour, so you'll have to multiply our CFM value by 8.

10.66 ft.3/Minute * 8 = 85.33 ft.3/Minute

So in this example, we'll need an exhaust fan with at least an 86 CFM rating.
Quick formula: (Length * Width * Height) / 7.5 = Required CFM rating.
Making sure the fan runs long enough
When installing an exhaust fan in a bathroom, it's a good idea to put the fan on a timer. This allows you to run the fan for the recommended 20-30 minutes after a shower, without having to worry about remembering to turn it off.
Warning: Make sure you get a timer that is rated for motor loads.