Alexander was born in Newark, New Jersey, the son of Jewish parents Ruth Minnie (née Simon), a nurse and health care administrator, and Alexander B. Greenspan, an accounting manager whose first name Jay later borrowed to create his stage name. He has a half-sister, Karen Van Horne, and a half-brother, Michael Greenspan. Alexander grew up in Livingston, New Jersey, and is a 1977 graduate of Livingston High School. He attended Boston University but left the summer before his senior year, after getting work in the city of New York. At Boston University, Alexander wanted to pursue classical acting, but a professor redirected him toward comedy after noticing his physique, remarking, "I know your heart and soul are Hamlet, but you will never play Hamlet. " He was awarded an honorary degree in 1995. He is a practicing magician, and switched to acting as a career only after deciding that he was unlikely to succeed professionally in a magic career.