Bain was born in Lethbridge, Alberta, the son of Jean Agnes (née Young) and Stafford Harrison Bain, who was a wholesaler. He studied at the Banff School of Fine Arts before serving in the Canadian Army during World War II. He then studied in New York at the American Academy of Dramatic Arts, where his classmates included actor Charles Durning and comedian Don Rickles; he became a naturalized citizen of the United States in 1946 before graduating in 1948.

Bain had three sons and a daughter with Monica Sloan, to whom he was married until her death in 2009. His identical twin brother is Bonar Bain, who once played the brother of his Maude character, Dr. Arthur Harmon.

After a stint at the Stratford (Ontario) Shakespeare Festival, Bain had further success as a stage actor in the 1956 Broadway revival of Eugene O'Neill's The Iceman Cometh. The New York Times reviewer noted that his role was "especially well acted." Bain's other Broadway theatre credits include Candide, Advise and Consent, An Enemy of the People, Uncle Vanya, and On Borrowed Time. On Off Broadway, he appeared in the original run of Steambath.While doing stage work in New York City, Bain also found work on television appearing in the soap opera Dark Shadows as the town inkeeper during season 1 and 2.

In the early 1970s, Bain appeared in New York-based films like Lovers and Other Strangers and Woody Allen's Bananas before achieving national recognition for his work in television. He is best known for his roles as Dr. Arthur Harmon in Maude (1972-78) and Phillip Drummond in Diff'rent Strokes (1978-86). In 1996, Bain reprised his role of Phillip Drummond on the series finale of the Fresh Prince of Bel Air.