Goldstein was born in Kingstree, South Carolina, the son of Fannie (Alpert) and Isadore E. Goldstein, who owned a clothing store. Goldstein received his BSci from Washington and Lee University in 1962, and his MD from Texas University's Southwestern Medical School in 1966. Upon completion of his residency, Goldstein moved to the National Institutes of Health (NIH) in Bethesda, Maryland, where he worked in biochemical genetics. In 1972, Goldstein relocated back to the Southwestern Medical Center, accepting a post as the head of the Division of Medical Genetics.