Conn was born in Brooklyn, New York, the daughter of a clinical psychologist. "Didi" was her childhood nickname. She attended Midwood High School; her brother is opera singer Richard Bernstein.

She has been married to composer David Shire since 1982, with whom she has a son, Daniel Shire, who has been diagnosed with autism. Conn is also stepmother to Matthew Shire, a Los Angeles screenwriter.

Conn made her debut as an actress in the 1960s. Her notable roles since the 1970s, when she first became prominent, include:

Conn also provided the voice for Raggedy Ann in the 1977 animated feature, Raggedy Ann & Andy: A Musical Adventure, directed by Richard Williams. Her "childlike" voice is one of her trademarks. Conn also appeared as a panelist a few times on the 1970s game show Match Game.

Conn's son Danny is diagnosed with autism. On November 13, 2008, Conn was named national celebrity spokesperson for Autism Speaks. Before that, Conn was a spokesperson for the National Alliance for Autism Research, now part of Autism Speaks. She has performed at benefits for Foundation for Education Children with Autism (FECA).

On September 27, 2008, Conn performed with David Shire and Lynne Wintersteller at a benefit performance for Barack Obama in Nyack, New York.