Stoltzman was born on July 12, 1942, in Omaha, Nebraska, but moved with his family to San Francisco at a young age; his father was a Western Pacific railroad worker who enjoyed the music of the big bands and played the alto saxophone. The young Stoltzman's first music was jazz. He took up the clarinet at eight and was soon joining his father in Bay Area jazz venues. Both Stoltzmans sang in a church choir where Richard would cue off-pitch singers with his clarinet, trying to make it sound like a voice. He later credited this experience for his distinctively warm tone quality. Stoltzman attended Ohio State University, studying mathematics and music, graduating in 1964. He chose the latter definitively, enrolling at Yale for graduate studies (earning a master's degree in 1967) and studied further at the Marlboro Music School and Festival. There he made the acquaintance of a group of collaborators that included pianist Peter Serkin, cellist Pablo Casals, and violinist Ida Kavafian. Together with Serkin and Kavafian, he formed the chamber group TASHI (a Tibetan term meaning good fortune) in 1973. Stoltzman was the first clarinetist to give solo recitals at both the Hollywood Bowl and Carnegie Hall, where he made his New York solo debut in 1974. His jazz career also developed during this period. In his concert appearances he joined an impressive variety of classic and modern performers including vocalist Mel Tormé, saxophonist Wayne Shorter, keyboardist Chick Corea, as well as folk-pop star Judy Collins. He was the first performer to feature jazz during a recital at the legendary Bayreuth Opera House. His classical collaborators have included cellist Yo-Yo Ma, pianist Christoph Eschenbach, and he has appeared as soloist with numerous major symphony orchestras in the U.S. and abroad. Stoltzman was featured in the series Concerto! on the BBC in 1993, performing Copland's Clarinet Concerto with the London Symphony Orchestra.
In 1986, Stoltzman became the first wind player to win the Avery Fisher Prize, given for outstanding performance by an American musician.
Beginning in 1982 with a classic recording of the Brahms clarinet sonatas, Stoltzman had a long association with the RCA label. During this time, he worked extensively with Bill Douglas (touring and recording together for 30 years) and released one of his most popular recordings, Begin Sweet World in 1986. Later he recorded for MMC, Navona, and, beginning in 2019, Avie, where he issued an album recorded jointly with his wife, jazz marimbist Mika Stoltzman, titled Palimpsest.