Saltonstall was born in Chestnut Hill, Massachusetts, and was a longtime summer resident of Vinalhaven, Maine. As an adult he spent winters on his family estate in Dover, Massachusetts, where he liked to farm. His father was Richard Middlecott Saltonstall, a lawyer; his mother, Eleanor Brooks Saltonstall, was the daughter of a multimillionaire, Peter C. Brooks. He married Alice Wesselhoeft (1893–1981) in 1916, and together they had six children, including Emily (1920–2006), at one time the daughter-in-law of Richard Byrd and a former WAVE; Peter Brooks Saltonstall, killed in action on Guam on August 13, 1944; William L. Saltonstall (1927–2009), a former member of the Massachusetts Senate; and Susan (1930–1994), a horse breeder.
A graduate of the private Noble and Greenough School, he graduated from Harvard College in 1914, where he was captain of the Junior Varsity crew that won the prestigious Grand Challenge Cup at the Henley Royal Regatta – the first American crew ever to do so–and Harvard Law School in 1917. Saltonstall also played hockey at Harvard where he scored a dramatic overtime goal in 1914 for Harvard to beat the legendary Hobey Baker's Princeton team. "Salty" also rowed and played football at Harvard.