She was particularly closely associated with baroque and early Italian opera and the works of Benjamin Britten. During her career, which spanned from the 1950s to the 1980s, she was considered an outstanding singing actress and widely admired for her dramatic intensity, perhaps best represented in her famous portrayal as Dido, the tragic heroine of Berlioz's magnum opus Les Troyens. As a concert performer, Baker was noted for her interpretations of Mahler and Elgar.

Baker was born in Hatfield, South Yorkshire. In 1956, she made her stage debut with the Oxford University Opera Club as Miss Róza in The Secret. That year, she also made her debut at Glyndebourne. In 1959, she sang Eduige in the Handel Opera Society's Rodelinda; other Handel roles included Ariodante (1964) and Orlando (1966), which she sang at the Barber Institute, Birmingham.

With the English Opera Group at Aldeburgh, Baker sang Purcell's Dido and Aeneas in 1962, Polly (Britten's version of The Beggar's Opera) and Lucretia. At Glyndebourne she appeared again as Dido (1966) and as Diana/Jupiter (Calisto) and Penelope (Il ritorno d'Ulisse). For Scottish Opera she sang Dorabella, Dido, Octavian, the Composer and Gluck's Orfeo ed Euridice.

In 1966, Baker made her debut as Hermia at the Royal Opera House, Covent Garden, and went on to sing Berlioz's Dido, Kate in Owen Wingrave, Mozart's Vitellia and Idamantes, Walton's Cressida and Gluck's Alceste (1981) there. For the English National Opera, she sang Poppaea, Donizetti's Mary Stuart, Charlotte (Werther) and the title role of Handel's Giulio Cesare.

In 1982 Baker retired from opera, after singing Mary Stuart at the ENO and Gluck's Orpheus at Glyndebourne. She published a memoir, Full Circle, in 1982. In 1991, Baker was elected Chancellor of the University of York. She held the post until 2004, when she was succeeded by Greg Dyke.