I don't know of any public launches on clearwater beach nor Sand Key (where the csc is) - http://www.bigpinellas.com/fish/boatramps.shtml

There's a boat ramp at the Seminole Boat Ramp about 1-2 miles north east of the CSC (on the mainland, not the clearwater beach / sand key) but it's not far at all and there are well marked channels to get you through and to the gulf (or CSC - but the waters can be very shallow around there)