Curtis was born in The Bronx, New York. His birth name is Bernard Schwatz. He is one of three sons of Helen (nee Klein) and Emanuel Schwartz. His parents were Hungarian Jewish immigrants from Mátészalka, Hungary. Hungarian was Curtis's only language until he was five or six, postponing his schooling. His father was a tailor and the family lived in the back of the shop—his parents in one corner and Curtis and his brothers Julius and Robert in another. His mother once made an appearance as a participant on the television show "You Bet Your Life", hosted by Groucho Marx. Curtis said, "When I was a child, Mom beat me up and was very aggressive and antagonistic." His mother was later diagnosed with schizophrenia. His brother Robert was also institutionalized with the same mental illness.