Barker studied with Dame Joan Hammond at the Victorian College of the Arts and began her career as a member of the opera chorus at the Victoria State Opera at the age of 19. She made her first solo appearance at that house as Blondchen in Mozart's Die Entführung aus dem Serail at the age of 23.
In 1984, Barker moved with her husband, baritone Peter Coleman-Wright, to London, so that he could assume his new position as a member of the choir at the Glyndebourne Festival. Two weeks after their arrival, Barker auditioned for and was accepted as a member of the opera chorus at the Welsh National Opera. She stayed there for the next year and a half, occasionally substituting for ailing singers in larger parts and performing the occasional comprimario role in addition to singing in the chorus.
In 1986, Barker was awarded the Dame Mabel Brookes Fellowship and was a finalist in the Metropolitan Opera National Council Auditions. These accolades helped her to launch a solo career with minor opera companies during the late 1980s. She performed the roles of Marzelline in Fidelio and Cherubino in The Marriage of Figaro with the English Touring Opera. In 1989 she won the Royal Over-Seas League Competition in London. Her first major critical success came in 1990 at her debut at the Sydney Opera House as Mimì in Puccini's La bohème, directed by Baz Luhrmann.
Since the early 1990s Barker has been a regular performer with all of the major opera companies in Australia. With the State Opera of South Australia she has sung Blondchen and Mimì. At the Victoria State Opera she has appeared as Antonia in The Tales of Hoffmann, the First Lady in The Magic Flute, and Mimì. She has portrayed Tatyana in Eugene Onegin at the Lyric Opera of Queensland and Cio-Cio San in Madama Butterfly with Auckland Opera. With Opera Australia she has performed Countess Almaviva in The Marriage of Figaro, Cio-Cio San, Desdemona in Otello, Emilia Marty in The Makropulos Affair, Lauretta in Gianni Schicchi, Marie/Mariette in Die tote Stadt, the Marschallin in Der Rosenkavalier, Marzelline in Fidelio, Nedda in Pagliacci, and the title roles in Arabella, Manon Lescaut, Salome and Suor Angelica.
In 2011, Barker sang the title role in Opera Queensland's production of Tosca, a role she had portrayed before in London and Sydney. After withdrawing from a 2010 production of Tosca by Opera Australia, she sang this role for the second half of Opera Australia's production in August 2013. Later that year, she reprised the role of Desdemona with Opera Queensland.
Barker's appearances with the English National Opera include Donna Elvira in Don Giovanni, the Foreign Princess in Rusalka, The Governess and Miss Jessel in The Turn of the Screw, Oksana in Christmas Eve, Musetta, Tosca, and the title role in Richard Strauss's Salome.
As a guest artist Barker has appeared at De Nederlandse Opera (Cio-Cio San, Emilia Marty), the Royal Opera, London (Jenifer in The Midsummer Marriage), the Scottish Opera (Adina in L'elisir d'amore, Annius in La clemenza di Tito, and Tatyana), the Vlaamse Opera (Suor Angelica, Liù in Turandot), and the Welsh National Opera (Káťa Kabanová). In 2004 she portrayed Sarah Miles in the world premiere of Jake Heggie's The End of the Affair at the Houston Grand Opera. In 2009 she sang Cio-Cio San for her first appearance with the Paris Opera.
The inaugural Australian cruise of the performing arts on the MS Radiance of the Seas in November 2014 included Barker as well as, among others, David Hobson, Colin Lane, Teddy Tahu Rhodes, Simon Tedeschi, Elaine Paige, Marina Prior, and Jonathon Welch. Barker and Hobson sang "O soave fanciulla" from La bohème, the first time they sang together 24 years after Baz Luhrmann's production.
Barker received the award for Best Female Performer in an Opera in the 2008 Helpmann Awards for her role in Arabella, and in 2013 for Salome.
In 2008, Barker and her husband were awarded honorary doctorates from the University of Melbourne. In the 2015 Queen's Birthday Honours, Barker was appointed an Officer of the Order of Australia, as was her husband.
Barker has one son, Gabriel, with her husband Peter Coleman-Wright. The couple has also recorded a CD of opera duets together with the Tasmanian Symphony Orchestra under conductor Martin André. The family maintains homes in London and Australia.