He is also known for portraying Peter in The Diary of Anne Frank (1959) and Ben Horne on the television series Twin Peaks (1990–1991, 2017).
Beymer was born in Avoca, Iowa, to George Richard Beymer, a printer, and his wife, Eunice (née Goss). He and his family moved in 1940 to Los Angeles, California.
In 1949 Beymer began acting in television in the TV series Sandy Dreams.
Beymer made his feature-film debut in Vittorio De Sica's Stazione Termini, released in 1953. He played in So Big (1953) and Johnny Tremain (1957), and achieved success in The Diary of Anne Frank (1959) and West Side Story (1961) before sharing a 1962 Golden Globe Award for New Star of the Year – Actor with Bobby Darin and Warren Beatty. He had a significant role in the film The Longest Day (1962).
In 1961, he began a relationship with a then unknown Sharon Tate, who was working as an extra on a film he was starring in, Hemingway's Adventures of a Young Man. Beymer encouraged her to pursue an acting career, and after she was introduced to his agent, Tate signed a contract with Filmways. He and Tate were engaged to be married for a year before they ended it. She went on to marry director Roman Polanski and was tragically murdered in 1969 by the Manson family. Richard has never spoken publicly about his relationship with Tate or any subsequent relationships after this event.
He took a summer away from Hollywood to volunteer in Mississippi for Freedom Summer canvassing for voter registration. He assisted Barney Frank in rescuing Freedom Democrat forms in a rental truck that had been confiscated from arrested Freedom volunteers in Canton, Mississippi on Freedom Day (July 16, 1964).
In 1957 he appeared on the syndicated American Civil War drama Gray Ghost in the episode "An Eye for an Eye". In 1958 he was cast as Joe Belden in the episode "Man Hunt" of the western aviation series, Sky King. He appeared in many guest roles in television series. These include three appearances on Star Trek: Deep Space Nine as Li Nalas in the episodes "The Homecoming", "The Circle", and "The Siege". In 1984 he played David Fenton, the husband of Mimi Rogers's character on the short-lived show Paper Dolls.
He had a role in the television series Twin Peaks in 1990, playing Ben Horne. He later played Dr. Matthew Sheridan in the 1996 made-for-TV movie A Face To Die For. Beymer reprised his role as Ben Horne in the third season of Twin Peaks in 2017.