Prisana, the floor in the closet is concrete with an epoxy coat. With the fan going constantly, the floor dries really quickly, so I don't worry too much about water on the floor. I intend, at some point, on putting up marine board to replace the sheetrock, in case mildews or molds decide to take hold, but I haven't had any problems yet.
Also, on the bottom of the 4' rack, I've got big plastic pans (masonry troughs) with my phals to catch any extra water. I use a pump sprayer and mist the pots until they drip through every day, but not very much drips through the racks.
I've got a half bath right across the hall from the closet, so it's relatively convenient if I want to soak all of the plants. I take one of the extra masonry troughs (used for mixing concrete) and load up a few, walk a few feet across the hall, set the tub on the toilet, and fill up the sink with the plants in it. Drain, carry, and repeat for all the others.
It's definitely not the easiest, but it's the about the best I can do in this house. I do miss the days of having a decent outdoor spot for everyone where I could just hit them with the hose everyday, though.
