Hafer began playing clarinet at age seven and switched to tenor sax in high school. His first professional gig was with Charlie Barnet's orchestra in 1949. He played with Claude Thornhill from 1949 to 1950 before returning briefly to play with Barnet again. After this he played with Woody Herman (1951–55), Tex Beneke (1955), Bobby Hackett (1957–58), Elliot Lawrence (1958–60), and Benny Goodman (1962). In 1963 he recorded on two Charles Mingus albums.
With Johnny Hartman
- The Voice That Is! (Impulse!, 1964)
With Charles Mingus
- Mingus Mingus Mingus Mingus Mingus (Impulse!, 1963)
- The Black Saint and the Sinner Lady (Impulse, 1963)
- "Dick Hafer Obituary". legacy.com. Retrieved 2012-12-25.
- Los Angeles Times obituary, December 22, 2012, page AA5.