To check for moisture, get some 6 mil plastic and carefully tape a 6 by 6 square on to the concrete floor. After a couple of days, if you see condensation on the plastic, then you have moisture coming through the slab. In that case, I would use some sort of barrier on the floor before I stored things near and dear to me in this space.
Regarding crawlspace vents, most current research shows that vented crawlspaces are not the best way to go. Go here:
RR-0401: Conditioned Crawl Space Construction, Performance and Codes — Building Science Information
for a pretty thorough discussion of this issue. This is a deceptively complicated issue, so if your current situation is working (no vents), I would not recommend venting your crawlspace. Depending upon where you live, you could make things worse, not better.