Leadfingers sent me a PM and pointed out that there is a link above (click) that will allow you to listen to the program (at least currently).

Here's the explanatory text from BBC:

Arthur Miller's recollection, in his 90th year, of an early job that was temporary but had a permanent effect on him.: recording the voices, the protest songs and the blues of striking black workers and railroad-men in North Carolina, just before the US entered the Second World War.

In 1941 Arthur Miller, penniless, got a job with the Library of Congress. He was sent South to record accents. Miller arrived in Wilmington, North Carolina a few weeks before Pearl Harbour. It was a critical time, and a critical place. In the former slave port a huge new shipbuilding facility had just been built to provide ships for the navy and Atlantic convoys. But the black people who had built the yards could not get work making the ships. There was also strife in textile manufacturing.

Miller was collecting accents but was more interested in what the people had to say rather than the way they said it.

Most striking to Miller and amazing to hear today were the people making music out of their experience and struggle - a railwayman singing raw blues and striking women shirt makers. So taken was he by this music he found a hall and recorded several songs and interviews.

Miller's recording trip only lasted a few weeks but it had a profound effect on him. He had never been to the South before and was shocked by the racism and anti-semitism. He was held at gunpoint for being Jewish. Once he made his host, a health care organiser, incandescent with rage by addressing a black man as 'sir' rather than 'boy' (the man was in his fifties, Miller in his early twenties).

Christopher Bigsby, the leading authority on Miller, discovered this material while working on Miller's diaries. At his house in Connecticut Bigsby played the material to Miller and interviewed him about this unknown but formative experience. So Arthur Miller tells the story of how, quite by chance, the playwright became a music collector.

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If the link above doesn't work, try this one (click).
BBC broadcasts tend to be available only for a short period of time.