I've found out that there is an article on this site about the photoshoot, in a series of articles about the history of computer chess. The chess set was rented from an antique store; it was a 16th century set from India, valued at $2,500, and the board was valued at $1,200. This was the photographer's idea, to make something impressive for a LIFE magazine article (that did not end up being published).
That article is
https://www.chess.com/blog/Ginger_GM/the-history-of-computer-chess-part-4-alex-bernstein-continued
and it also notes that Edward Lasker is the chess player in the photograph.
On YouTube, there is a newsreel of Dr. Alex Bernstein playing chess against his program running on an IBM 704 computer. In that film, the chess set used is a normal Staunton chess set.
In the volume Mathematics from the Life Science Library, though, there is a photo of Dr. Bernstein with another individual having the IBM 704 solve chess problems. This photo was taken by the noted photographer Andreas Feininger, and there are even prints of it for sale online.
In that photo, the set is an ornate set, appearing to be made of ivory and long predating the adoption of the Staunton pattern. It looked a lot like a Barleycorn type set, but there were some differences, particularly in the appearance of the Rooks.
Here is that photo as an inset in a detail from the photo, large enough so you can see the kind of chess set that is being used here.
Hunting around the Internet, in a 1978 Christies' catalog on-line, I saw a similar looking set billed as an Indian Ivory and Horn chess set.
With that clue to narrow my search, I've found at least one set of that general type for sale online:
https://chessantiques.com/product/antique-vizagapatam-chess-set/
The Rooks don't have flags, but they have a "mushroom" or "hat", and they have interlocking brickwork, which was the distinguishing feature. And this set is also from ivory and buffalo horn, like the one sold by Christies.