You're being thrown off by a poor choice of names that Drinfel'd made. (Hey, he had to name a lot of stuff, so I can't really blame him for getting one wrong.) By "unitarization" he means "modifying it so that it's square is 1" which has nothing whatsoever to do with being unitary. The usual R-matrix should be unitary wrt to the usual *-structures on quantum groups if q is a number of the right form (either size 1 or real depending on the *-structure).
Edit: As pointed out in the comments I seem to be wrong here. I'm going to leave up the answer unmodified, however, because the comments clarify the situation and editing it would make the comments more confusing.
