Aldredge carved out a respected career on the Broadway stage that spanned five decades. He made his Broadway debut as Danny in the 1959 musical The Nervous Set. In 1972 he won a Drama Desk Award for his portrayal of Ozzie, the father of a blinded Vietnam veteran, in David Rabe's Sticks and Bones. He played Henry VIII's fool, Will Sommers, in Richard Rodgers' pentultimate musical, Rex in 1976. He originated the role of Norman Thayer Jr. in On Golden Pond in 1978, earning the first of three Tony Award nominations. His best-known role, however, was that of the Narrator/Mysterious Man in Stephen Sondheim and James Lapine's Into The Woods, a role he later repeated in the PBS Great Performances production. He also created the role of the Doctor in another Sondheim/Lapine collaboration, Passion.