Daly was born March 1, 1956, at Mount Sinai Hospital in New York City, the only son and youngest of four children of actor James Daly and actress Mary Hope (née Newell) Daly. He is the younger brother of actress Tyne Daly. He has two other sisters, Mary Glynn (wife of Mark Snow) and Pegeen Michael. Daly attended The Putney School, where he started to study acting.
Daly began his professional career while a student at Vermont's Bennington College, where he studied Theatre and Literature, in which he now holds a Bachelor of Arts degree, and acted in summer stock. He graduated from college in 1979, and returned to New York to continue studying acting and singing.
Daly debuted on stage when he was seven years old in Jenny Kissed Me by Jean Kerr, together with his parents and two sisters. He appeared for the first time on TV when he was 10 years old in an American Playhouse adaptation of An Enemy of the People by Henrik Ibsen, which starred his father James Daly. He dreamed about a sports or music career and also considered becoming a doctor or a lawyer, but finally decided to become an actor. Daly started his professional acting career when he appeared in a 1978 adaptation of Peter Shaffer's play Equus.
His first leading film role was in the film Diner, directed by Barry Levinson, in which he shared screen time with actors including Kevin Bacon and Mickey Rourke. Starring roles soon followed in Alan Rudolph's feature, Made in Heaven, the American Playhouse production of The Rise & Rise of Daniel Rocket, and the CBS dramatic series, Almost Grown created by David Chase.
In theatre he has starred in the Broadway production of Coastal Disturbances by playwright Tina Howe opposite Annette Bening and received a 1987 Theatre World Award for his performance. He has also starred in Oliver, Oliver at the Manhattan Theatre Club, Mass Appeal by Bill C. Davis and Bus Stop by William Inge at Trinity Square Repertory, The Glass Menagerie by Tennessee Williams at the Santa Fe Festival Theatre, A Knife in the Heart and A Study in Scarlet at the Williamstown Playhouse, and Paris Bound at the Berkshire Theatre Festival. During this time, Daly also starred in the CBS television miniseries I'll Take Manhattan as Toby Amberville.
Daly describes himself as being highly self-critical in regard to his career. In an interview with New Zealand 'ZM' radio personality Polly Gillespie, Daly was quoted as saying, "I think part of it (his self-critical nature) is passed down to me from my parents who are actors. The theatre was our temple... When you entered you were expected to live up to the example of this glorious place.