Born in Antwerp, Grosbard was the son of Rose (Tenenbaum) and Morris Grosbard, who worked in Business and as a Diamond merchant. Grosbard emigrated to Havana with his family in 1942; they were fleeing the persecution of Jews by the German occupiers of Belgium during World War II. In 1948, they moved to the United States, where he earned Bachelor of Arts and Master of Arts degrees from the University of Chicago. He studied then at the Yale School of Drama for one year before joining the United States Army. Grosbard became a naturalized citizen in 1954.