You may recall the expository cartoon, explaining the principles of spaceflight, embedded within the film. In it, Woody Woodpecker encounters an issue of Life covering the movie he is in. In reality, the April 24, 1950 issue carried a Destination Moon feature, though the cover showed a girl, not a Moon rocket.
Life sent Allan Grant to shoot the production. A few of the photos have captions (I presume these are the ones which appeared in the magazine story) but most of them do not. Fortunately-- though these were the days before DVD extras and "making-of" documentaries were commonplace-- Heinlein wrote a magazine article about the production, which helps in understanding some of Grant's photos. The full text of "Shooting Destination Moon" is not online, but you can find it in the book Requiem, among other places.
Checking the callsheets in the UC Santa Cruz archive of Heinlein's papers, I believe Grant shot these during the first two weeks of December 1949.
A few gag shots crept in. Some are just bizarre.
I'm afraid I can't tell you who these people are. They don't appear in the movie.
Rigging a crewman for a spacewalk scene.
Life:"Actors in space suits outside rocket ship as they use Geiger Counter to look for radioactive material while midget actors in space suits work in bkgrd. because their smaller size gives illusion of distance."
Life:"MIDGET ACTORS are helped by stagehands over rough terrain. Actor in front was hung by wire so he could make high leaps as he ran on moon's surface where power of gravity is less. This picture was taken at same spot as one on previous page, but small-scale rocket was put in rear to create a different view."
Nice view of the overhead cranes, piano wires, and netting used to keep stuntmen aloft for the spacewalk rescue scenes.
I really don't understand what's going on here. Write your own caption!
If you're wondering about some of the sets, special effects, props, and costumes shown in these photos, William Max Miller has a good discussion on "The Filming of Destination Moon."
A few more behind-the-scenes photos illustrate an article on Moon rockets in Popular Mechanics, May 1950.
Full-page ad in PopMech for the movie, September 1950.