is it a fiberglass / acrylic / or marble base with a drian like is pictured in Old House Plumber's post? pop the strainer off and see how it is sealed between the pipe and drain housing.
is the inside of the drain housing threaded or smooth / can you see the top of a gasket between the pipe and housing / if threaded is there a "castle" nut or some kind of nut tightened down on top of the gasket? Is the drain housing brass or plastic / if plastic, is there a space for a gasket or is it a solvent weld (glue) joint between the pipe and drain housing? Can you see the drain leaking / sure that it is leaking between the pipe and drain housing or could it be leaking between the drain housing and the shower base?