No they have not. The closest they've ever gotten to winning it all was in 1964 when they made it to the Elite Eight.
Ohio State.
4
Ohio State defeated California 75-55 in the 1960 NCAA men's basketball tournament.
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NCAA Championship 1960 OSU They Have Won One
Four: Xavier, Ohio State, Ohio University and Cincinnati University
Ohio State University
This year, the 2012 NCAA Tournament has four teams from Ohio in the Sweet 16: The Ohio State University (Go Bucks!), Ohio, Cincinnati, and Xavier.
1939 was the first NCAA tournament. The teams participating were: Oregon, Texas, Oklahoma, Utah State, Villanova, Brown, Wake Forest, Ohio State Ohio State and Oregon played in the Final with Oregon winning the first NCAA Tournament.
2006. Florida defeated Ohio State in both the BCS National Championship Football game and the NCAA Basketball Tournament.
Oregon won the 1939 NCAA tournament (the first NCAA tournament, mind you) with a 46-33 victory over Ohio State. They beat Texas and Oklahoma to reach the final of the 8-team tournament.
There was no 1933 NCAA Tournament. The first NCAA Tournament was in 1939 with Oregon beating Ohio State in the Finals.