If you have to stay at the airport I'd strongly suggest one of the hotels just south of it as at least you have a bit of a water view. Just checked and the Hilton on the Blue Lagoon is going for USD 104 per night starting May 24th. That's not bad at all.
Since May is the end of the shoulder season, beginning of the off season down there, you should be able to find quite a few comparable deals.
Coral Gables? Not too far from the airport and most parts seem to be a world away from Miami. Great area.