McCloskey was classically trained as an actor at the Juilliard School in Lincoln Center in New York City. He began his professional acting career playing Billy Abbott in the original miniseries Rich Man, Poor Man. He went on to star in three nighttime television series: Executive Suite, Married: The First Year, and performed the role of Mitch Cooper (husband of Lucy Ewing Cooper) on the CBS nighttime soap opera Dallas. McCloskey was a series regular from 1980–1982, and made brief appearances in 1985 and 1988.