Douglas was born Issur Danielovitch on December 9, 1916, in Amsterdam, New York. He was the son of Bryna "Bertha" (née Sanglel; 1884–1958) and Herschel "Harry" Danielovitch (c. 1884–1950). His parents were Jewish emigrants from Chavusy, Mogilev Region, in the Russian Empire (present-day Belarus), and the family spoke Yiddish at home.
His father's brother, who emigrated earlier, used the surname Demsky, which Douglas's family adopted in the United States.
Growing up, Douglas sold snacks to mill workers to earn enough to buy milk and bread to help his family. Later, he delivered newspapers and during his youth worked at more than forty different jobs before getting a job acting.