When you say "damp", what do you mean? Where is this damp coming from?
Sounds to me like you probably have about 4",(if you are lucky), of cement between You and the dirt below the garage floor.
You should have installed a vapor barrier over the cement before any walls or closets were installed. (anything stored in the closet will probably mold and mildew)
You say this is a step down, how much headroom do you have?
I may have used pressure treated 2 X 4s, on the flat, (if the you are short on headroom,) to lay out the floor and used 1 1/2" high density foam insulation between the 2 x's to make the vapor barrier. This would also make the floor warmer and the room easier to heat.
You don't want to make the mistake of just installing carpet to make the floor warmer, all you will be doing is installing a sponge.
When you built the walls, did you use P.T. wood where the walls meet the cement?
I know you probably do not want to read this but, in order to do this right, you may have to rebuild the walls.
Note* Most garages are pitched so water will run out. By installing a subfloor, you will be able to level the floor.