Boasberg was born in San Francisco, California in 1963, to Sarah Margaret (Szold) and Emanuel Boasberg III. The family moved to Washington, D.C. when Boasberg's father accepted a position in Sargent Shriver's Office of Economic Opportunity, a Great Society agency responsible for implementing and administering many of Lyndon B. Johnson's War on Poverty programs. Boasberg received an Bachelor of Arts from Yale University in 1985, where he was a member of Skull and Bones, and a Master of Studies the following year from Oxford University. He then earned his Juris Doctor from Yale Law School in 1990.
After completing law school, Boasberg served as a law clerk for Judge Dorothy W. Nelson of the United States Court of Appeals for the Ninth Circuit for the 1990-91 term. He then went into private practice, working at firms first in San Francisco (1991-94) and then in the District of Columbia (1995-96). In 1996, Boasberg joined the office of the United States Attorney for the District of Columbia where he would spend 5.5 years as a prosecutor, specializing in homicides.
In September 2002, Boasberg became an Associate Judge of the Superior Court of the District of Columbia, where he served in the Civil and Criminal Divisions and the Domestic Violence Branch until his appointment to the federal bench in 2011. During the 111th Congress, Delegate Eleanor Holmes Norton recommended Boasberg to fill a judicial vacancy on the United States District Court for the District of Columbia. On June 17, 2010, President Barack Obama formally nominated Boasberg to the District Court for the District of Columbia. Boasberg was confirmed on March 14, 2011 by a vote of 96 ayes to 0 nays. He received his commission on March 17, 2011.
On April 26, 2012, Judge Boasberg ruled that the public had no right to view government photos of a deceased Osama Bin Laden. Judicial Watch, a conservative legal group, had filed a request under the Freedom of Information Act (FOIA), but were unsuccessful in convincing the Judge that FOIA rights outweighed national-security factors.
On January 18, 2013, Judge Boasberg issued a permanent injunction prohibiting the Internal Revenue Service from enforcing regulations on Registered Tax Return Preparers, which otherwise required tax return preparers to register with the IRS and pass a written test as evidence of competency. Loving v. Internal Revenue Service, No. 12-385 (U.S.D.C. D.C. 1/18/2013). The IRS appealled and in 2014, the Court of Appeals upheld Boasberg's district court decision.
On February 7, 2014 Chief Justice John G. Roberts announced that he would appoint Boasberg to the United States Foreign Intelligence Surveillance Court for a term starting May 18, 2014 to a seat being vacated by Reggie Walton. His term began May 18, 2014.
On August 22, 2016, Boasberg ordered the release of over 14,000 emails found in the United States Department of State correspondence of Hillary Clinton by the FBI during an investigation of Clinton's private server. These emails were requested by Judicial Watch, a conservative legal group, because the FBI has indicated that emails were work-related and not entirely private as Clinton had previously said.
On August 18, 2017, Boasberg dismissed a lawsuit from the Electronic Privacy Information Center (EPIC), which had sued the IRS under FOIA seeking for President Donald Trump's personal tax returns from 2010 to the present to be released. Boasberg concluded that because personal tax returns are confidential, they may only be obtained either by permission from Trump himself or if Congress's Joint Committee on Taxation signed off to allow the disclosure.
Boasberg married Elizabeth Leslie Manson in 1991. His brother, Tom Boasberg, succeeded Michael Bennet as Superintendent of Denver Public Schools after Colorado Governor Bill Ritter appointed Bennet to the United States Senate in January 2009.