Clarkson was born in New Orleans, Louisiana, the daughter of Jackie Clarkson (née Brechtel), a New Orleans politician and councilwoman, and Arthur Clarkson, a school administrator who worked at the Louisiana State University School of Medicine. She has four sisters. Clarkson studied speech pathology at Louisiana State University, later transferring to Fordham University in New York City, where she graduated summa cum laude with a degree in drama. She then earned her Master of Fine Arts at the Yale School of Drama before making her film debut in The Untouchables in 1987.
Clarkson's Broadway theatre credits include The House of Blue Leaves and Eastern Standard. She had supporting roles in a series of high-profile films in her early career. Her first movie role, at age 27, was as the wife of Eliot Ness (Kevin Costner) in The Untouchables. She was also featured in The Dead Pool, Rocket Gibraltar, and Everybody's All-American. She starred in the short-run television series Davis Rules, and in the miniseries Alex Haley's Queen.
More often a character actor than a leading woman, she has also appeared in such major films as The Green Mile (1999), Far from Heaven (2002), Good Night, and Good Luck (2005), and Lars and the Real Girl (2007). Her breakthrough role was in High Art (1998) where her portrayal of Greta, a German, drug-addicted lesbian former model, gained rave reviews. She has said about acting, "I’m deeply invested in everything I do, and it’s a good thing, because acting is the only thing I know how to do."
In 2002, she originated her role as Sarah O'Connor ("Aunt Sarah") in HBO's Six Feet Under. She went on to win the Primetime Emmy Award for Outstanding Guest Actress in a Drama Series that year for her performance; she repeated for the same role in 2006. In 2003, she gained critical acclaim and a Special Jury Prize for her work in four films that debuted at the Sundance Film Festival: All the Real Girls, The Baroness and the Pig, The Station Agent and Pieces of April. Later that same year, she was nominated for an Academy Award for Best Supporting Actress for Pieces of April, in which she played an acerbic mother dying of cancer. In 2006, she portrayed Sadie Burke, in All the King's Men, set in her native New Orleans. In 2008, she had supporting roles in Elegy, as a womanizer's put-upon girlfriend, and Woody Allen's Vicky Cristina Barcelona as an unhappily married housewife.
In 2008, producer Gerald Peary approached Clarkson to do the voice-over for the documentary film For the Love of Movies: The Story of American Film Criticism. Says Peary, "She agreed to do the narration ... And she was so nice, and so cooperative, and so prepared, and so intelligent. And one of the key reasons she wanted to do the movie was that she regularly reads criticism, and has a genuine respect for the critic profession.
Clarkson returned to New Orleans for the January 17, 2009, reopening of the Mahalia Jackson Theatre for the Performing Arts. She served as master of ceremonies for a gala featuring Plácido Domingo in concert with the New Orleans Opera, conducted by Robert Lyall. She also made a cameo appearance in the Saturday Night Live Digital Short "Motherlover" on May 9, 2009. The video featured Andy Samberg, Justin Timberlake, and Susan Sarandon. She reprised the role on May 21, 2011, in the digital short "3-Way (The Golden Rule)". In 2010, she starred in Martin Scorsese's Shutter Island. Clarkson was honored by the Jameson Dublin International Film Festival when she received one of the 2010 Volta awards for achievements in her career.
In the fourth season of the NBC mockumentary Parks and Recreation, Clarkson began portraying the recurring role of Tammy One, Parks Department head Ron Swanson's first wife.
In response to the Deepwater Horizon oil spill, Clarkson published a post for Natural Resources Defense Council's magazine OnEarth. She also released a public service announcement talking about her experiences growing up in New Orleans. Both pieces were released on July 26, 2010.
Clarkson has never married and has no children. Interviewed in 2016, she said, "I've never wanted to marry, I've never wanted children – I was born without that gene."