Craft was born in Kingston, New York and studied music at the Juilliard School. He became particlarly interested in early music by the likes of Claudio Monteverdi and Heinrich Schütz, and contemporary music by the composers of the Second Viennese School and others. He is particularly noted for his close friendship with Igor Stravinsky, a relationship which has resulted in a number of recordings and books.
Craft met Stravinsky in 1948, from them until the death of the composer in 1971, Craft continued to work along side Stravinsky. A relationship that started as out as a meager role for Craft evolved into a full blown artistic partnership. Craft was the librettist for, 'The Flood'. Craft lived with the Stravinsky family in California and later in New York. Craft continued to remain close to the composer's widow Vera until her death in 1982.
Long after Stravinsky's death, Craft continues to concertize and to write. His most recent book, An Improbable Life, details his life before, during and after Stravinsky. The memoir elaborates on the impact that Igor Stravinsky has had and continues to have on his life.