The Scotsman, who won 13 grands prix in a Formula 1 career that spanned from 1994 to 2008, will take over from Paddy Hopkirk after he had been in the role since 2017.
Coulthard had been widely tipped for the role, which he will fulfil alongside his current broadcasting commitments for Channel 4 and his involvement with the new W Series.
John Grant, Chairman of the BRDC, said: "I and my fellow board Members are delighted to welcome David Coulthard as president of the BRDC.
"We are confident that, in David, we have a president who is not only a greatly talented racing driver but also a successful businessman with an enviable network of contacts in the motorsport world which, together with his experience, should be of great benefit to the club."
Coulthard was elected at this week's BRDC AGM, which also confirmed that Dario Franchitti would become a vice-president of the club.