Your concerns about possible future moisture is well-founded. And be sure that whatever you do, consider NOT covering over the drain. In fact, you might want to test it to be sure it's free and clear (and does what it's suppose to do.) If your washer is down there, you really do want it to work! (Oh, have you changed your 2 washer hoses recently??!)
Anyways, ... Have you considered commercial berber carpeting?