Galbraith moved to New York City from Vermont in the early 1940s and found work playing with Babe Russin, Art Tatum, Red Norvo, Hal McIntyre, and Teddy Powell. He played with Claude Thornhill in 1941-1942 and again in 1946-1949 after serving in the Army. He did a tour with Stan Kenton in 1953.
Galbraith did extensive work as a studio musician for National Broadcasting Company and Columbia Broadcasting System in the 1950s and 1960s.
Among those he played with were Miles Davis, Michel Legrand, Tal Farlow, Coleman Hawkins, John Lewis, Hal McKusick, Oscar Peterson, Max Roach, George Russell, John Carisi, and Tony Scott. He also accompanied the singers Anita O"Day, Chris Connor, Billie Holiday, Helen Merrill, Sarah Vaughan and Dinah Washington on record.
He was a mentor to Ralph Patt. In 1961 he appeared in the film After Hours.
From 1970 to 1975 he taught at City University of New York, and published a guitar method book in 1982.
From 1976-1977 Galbraith taught guitar at New England Conservatory in Boston.