Born Jane Quigley in Boston, Massachusetts, her parents were Bart Quigley (who was of Irish and German descent) and Ruth Pearson, whose mother was born in Nova Scotia, Canada. Alexander then studied at Sarah Lawrence College and the University of Edinburgh before beginning to act on the stage in Washington, DC.

She performed for several years at the Shakespeare Theater in Stratford Connecticut (including a fine performance in the title role of Major Barbara). In 1969 she debuted on Broadway in The Great White Hope for which she won a Tony Award as Supporting or Featured Actress - Play and took it to Hollywood, where she was nominated for the Academy Award for Best Actress. She received other nominations for All the President's Men (1976) and Kramer vs. Kramer (1979). She has starred in the television productions of Eleanor and Franklin and Eleanor and Franklin: The White House Years. Other television movies include Playing for Time, Calamity Jane, Malice in Wonderland, Blood & Orchids, In Love and War, and Daughter of the Streets. In 1983, she received another Oscar nomination for the post-nuclear war film Testament which co-starred William Devane.

In 1993, President Bill Clinton appointed her chairman of the National Endowment for the Arts. She served until 1997.

In 2005 she won an Emmy Award for Best Supporting Actress in a TV Movie/Miniseries, for her role as Sara Roosevelt in HBO's Warm Springs.