Baker appeared on Broadway in 1959 in Look After Lulu. Among his other stage credits are Aren't We All? in the West End; The Merry Wives of Windsor, The Importance of Being Earnest, and Saint Joan at the Old Vic; and Hamlet at the Royal Shakespeare Company. In addition, he was head of the Theatre Royal in Bury St. Edmunds from the mid-1960s to early 1970s.
Baker appeared in such films as Tread Softly Stranger, Goodbye Mr. Chips, and On Her Majesty's Secret Service, and The Spy Who Loved Me, and may be best remembered for his work as Tiberius in I, Claudius, and for playing Chief Inspector Wexford on television's The Ruth Rendell Mysteries.
He is survived by four daughters from his first marriage and one from his second.