Alexander Kossoff was born in the East End of London and educated at Dame Alice Owen's School at Islington. In 1926 he won a scholarship to the Royal Academy of Dramatic Art, where he Anglicized his name to Alan Keith. He graduated in 1928 with the Silver Medal, and spent the next eight years on the West End and Broadway stage.

By 1935, Alan Keith was already an established voice on BBC radio, appearing in dozens of radio plays as a member of the drama stock company and spending three years as an interviewer for In Town Tonight. He also acted in films, appearing in Dangerous Moonlight (1941). In pre-war television broadcasts, he discovered he had a facility with American accents, and he continued to play American characters on television and radio through the 1940s and 1950s.

Beginning in the early 1950s, he devoted time to devising and presenting music programmes for the BBC. In 1959, he devised Your Hundred Best Tunes, a programme of light classical music, operetta and ballads. Keith chose the original 100 pieces himself, but in subsequent years they were periodically voted on by listeners.

In early March, 2003, at the age of 94, he recorded an announcement that he intended to retire from the programme after 44 years. However, he fell ill almost immediately afterwards, and died soon after; his final programme was broadcast 12 days after his death.

He was the elder brother of the actor, David Kossoff.