He then received a Master's in Fine Arts from the Yale School of Drama, where he studied alongside Chris Noth and Patricia Clarkson.

[1] Baker's Broadway theatre credits include Eastern Standard, La Bête (for which he received a Tony Award nomination), Mauritius, and God of Carnage.

In 2000, he portrayed Secretary of Defense Robert McNamara in Thirteen Days, a historical drama about the Cuban Missile Crisis.

He also held a small role in Requiem for a Dream, where he played a doctor who sees Jared Leto's character with a severely infected, rotting arm.

For his performance as Colin Sweeney on The Good Wife, he was nominated for the Primetime Emmy Award for Outstanding Guest Actor in a Drama Series in 2010, 2012, and 2014.

[2] Baker performed with Helen Mirren on Broadway in The Audience, a play that depicted meetings between Queen Elizabeth II and important historical figures.

In 2016, Baker had a memorable recurring role as deep-cover KGB agent William Crandall, a morally conflicted biological weapons expert, on Season 4 of FX series The Americans.

[4] In September 2017, it was announced that Baker would be joining the cast of the Showtime drama Homeland for the show's 7th season, as Sen. Sam Paley, a 'maverick' who is leading an excessive investigation of the administration of new President Elizabeth Keane.

On September 1, 2015, Baker tried to save the life of his neighbor, Broadway and movie actress and dancer Vivien Eng, after her apartment caught fire in the New York City high-rise where they both lived at the time.