Clapp earned the B.M degree from the Oberlin Conservatory and the M.S. degree from the Juilliard School. He was a student of Dorothy DeLay, Ivan Galamian, and Andor Toth. He studied chamber music with Claus Adam, Robert Mann, Felix Galimir, Raphael Hillyer, Louis Persinger, and Walter Trampler.
As a member of the Beaux-Arts String Quartet, Clapp won the first Naumburg Chamber Music Award. He won the Josef Gingold Prize of the Cleveland Society for Strings while a student at the Oberlin Conservatory. Clapp was concertmaster of the Aspen Chamber Symphony, Nashville Symphony, and the Austin Symphony Orchestra.
Clapp performed in numerous summer festivals in Europe and North America. He was first violinist in the Blair String Quartet from 1967-72.
On April 28, 1985, Tim Page (music critic) of the New York Times had this to say about Clapps's playing in a New York debut recital: "Mr. Clapp, who was ably accompanied by the pianist Frances Walker, plays sweetly and songfully, with innate taste; he produces a dark, affecting tone from his instrument, which sounded almost like a viola at times. . . . a vigorous interpretation, which it received Sunday."
Clapp taught violin at a number of schools.
- Aspen Music Festival and School, 1971–94
- Peabody College, Nashville, TN), 1967–72
- University of Texas at Austin, TX, 1972–79
- Oberlin Conservatory, 1978–90
- The Juilliard School, 1987–2014
- Leon Kirchner Piano Trio, performed by the Oberlin Trio
- 20th-century American Piano Trios 
- French Trios - Maurice Ravel, Jean Baptiste Loeillet of Ghent, Claude Debussy
- SAD NEWS | Juilliard Former Dean Stephen Clapp Has Passed Away – Aged 74
- "Music Sales Classical". Schirmer.com. Retrieved 2014-06-18.
- "the pno Oberlin Trio - Twentieth Century American Piano Trios / The Oberlin Trio CD Album". Cduniverse.com. 2000-06-26. Retrieved 2014-06-18.
- "French Trios - Ravel, Loeillet, Debussy / Oberlin Trio CD Album". Cduniverse.com. 2000-06-26. Retrieved 2014-06-18.