Gluing them down is unlikely to solve the water problem.

There are a few ways you can work your issue. You can in fact glue them down. If you have BLT mats, you should use their Pressure Sensitive Adhesive. This is what they say to use. You can caulk around the edge to seal it, but if you have any expansion or contraction with your mats you might get bubbles in the mats. Or if you get a bunch of water you could get containment mat rather that the mats that you have right now.
This glue is made by BLT specifically for the mats
You can get the glue at http://www.jnkproducts.com/blt-adhesive.htm

Or the containment mats at http://www.garagefloor.com/park-smart-clean-park-special.htm