Concrete basement floors are generally more aesthetically pleasing and more comfortable if they are carpeted. Carpet lends warmth under your feet and to children sitting on the floor while playing or watching television. Carpet will also help deaden sound. The type of carpet you choose to use will be determined by what the room will be used to do.
If it's a game or party room, a short pile would be easier to clean and more durable. For warmth, a longer pile or plush carpet would be best. This type of carpet would also be better if you wish to deaden sound in a home theater room.