The simple summary is that the Caymans aren't Cozumel.
The currents are generally VERY light to non-existent on LC and CB except when the wind is blowing strongly, and then they just dive the leeward side. On LC and Brac, the boat captain and DM will check the current at the site before setting up for the dive and if it is even moderate, they will usually move to another location. I have never done a drift dive on LC or CB in 5 trips there.
I've had perhaps 3-4 drifts over the past 20 years; figure it to be <1% of my Cayman dives. However, this is not to say that there's never been any current.
A lot of this question depends on how strong/weak a current has to be until there's a problem. In 500+ dives in the Caymans, I've never been blown downcurrent, but I have seen a few divers who have. In general, I'd say that there's been two basic reasons why.
The first is navigation/dive planning ... they got lost, and too far downcurrent to be able to swim back without going OOA, so they surface downcurrent (and get picked up). Novices tend to stay close and verify where the diveboat is well before they're low on air, so this usually befalls Intermediate divers.
The second one happens usually just on wall dives ... there's a surface current, which doesn't become particularly evident until ascent to the safety stop. If you can't make it to the mooring line to hold on, you can get blown downcurrent. IMO, it's usually the Intermediate divers (again) who get hit with this one, since Novices will usually prefer to come up the mooring line, and Advanced divers who would otherwise do a midwater stop tend to "notice" earlier, so they'll dip back down to the bottom to get out of the current to swim in before ascending.
-hh