What we now call Old-Time Radio or the Golden Age of Radio was not old at all by the time Isaac Asimov became famous for his Science Fiction short stories. Science Fiction radio drama became popular in the early 1950s (or even a bit earlies if you count in Superman), and a number of radio shows featured stories by Asimov.
I try to list all shows here to the best of my knowledge and welcome any corrections or additions.
[1950-1951] Dimension X
Dimension X was a 50 episode NBC radio program. The first 13 episodes were aired live, then the format was switched to pre-recorded shows.Two episodes were based on Asimov stories and adapted by Ernest Kinoy:
An OTRR certified complete version of the series can be downloaded for free from archive.org.
[1955-1958] X Minus One
X Minus One was a 127 episode NBC radio program, which was very similar to its predecessor Dimension X. Three of the episodes were based on stories by Isaac Asimov:
Here again, a OTRR certified complete version of the shows can be downloaded at archive.org.
[1957-1958] Exploring Tomorrow
Exploring Tomorrow was a 29 episode series narrated by John W. Campbell and aired on the Mutual Broadcasting System. One episode was based on an Asimov story:
An mp3 file of this episode is available from archive.org.
[1973] The Foundation Trilogy (BBC)
In 1973, BBC produced an eight episode version of the Foundation Trilogy which was broadcast between 6th May and 24th June 1973.
According to archive.org, the series is in the public domain and can be downloaded there. I however prefer my eight tape boxed set from the BBC.
[1989] The Caves of Steel (BBC)
In 1989, BBC 4 broadcast a three episode version of The Caves of Steel with Ed Bishop (known as Ed Straker from the 1970s TV series U.F.O.) as Elijah Baley.
One might argue that podcasts are the radio shows of the 21st century.
There is actually one podcast, Episode 100 of Escape Pod, which features a dramatization of Asimovs Nightfall.