József Babán began his ice hockey career as a youth player in 1952 before making his senior début for BVSC in 1956 where he remained until 1969. From 1958 to 1965 Babán also played in 55 matches for the Hungarian national team where he competed at several World championships and the 1964 Innsbruck Olympics. While still an active player he coached the BVSC youth team from 1958 until 1968 before becoming the coach of the senior team. He worked with several other clubs and was also the assistant coach of the national team from 1979 to 1980. In 2013 Babán was inducted into the Hungarian Ice Hockey Hall of Fame.