Born in New Jersey, Mensch studied at the Manhattan School of Music and the Jacques Dalcroze School of Music; his teachers included New York Philharmonic principal bassist Anselme Fortier. As a teenager, he performed in the Dick Messner Big Band at the Hotel McAlpin in New York.
"This was at the start of the Great Depression," he told the Juilliard Journal recently, "so pursuing a career in music was pretty risky. I was lucky that my parents didn't try to make me go into a field that was a safer bet."
Mensch joined the Pittsburgh Symphony in 1932—after auditioning for Otto Klemperer on the stage of Carnegie Hall—and served as assistant principal bassist in 1937-38. He joined the Philharmonic in 1938, when the orchestra was led by John Barbirolli.
In 1943, Mensch left the Philharmonic to serve in the Army in Texas. A year later, he returned to New York as a freelance musician, performing with the NBC Symphony and for television and radio shows.
Mensch returned to the Philharmonic in 1966, during Leonard Bernstein's tenure, and remained through 1975. After leaving, he played with such orchestras as the Mostly Mozart Festival Orchestra, New York Chamber Symphony, and New York Pops.
He appeared on many classical recordings and did session work for movie soundtracks, commercials, TV shows, and pop music.
Mensch joined the Juilliard faculty in 1970 and became chair of the double bass department in 2002; he also taught at Yale, the Manhattan School of Music, Mannes College of Music, and other institutions. In addition to classical bassists, his former students include the jazz standouts Christian McBride and Steve Kirby.