Boswell was son of James Boswell (ca. 1672–1749), 7th Laird of Auchinleck. As a judge he took the courtesy title Lord Auchinleck from the name of his estate at Auchinleck House, of which he was the 8th Laird. He is noted for rescuing of the Auchinleck manuscript in 1740 from a professor of the University of Aberdeen who was in the process of destroying it. He donated the manuscript to the Advocates' Library in Edinburgh.
He is known for frequently arguing with his son, James Boswell, upon matters such as Boswell's career choice, his publishing methods and routes of travel, his friends and even his choice of wife.