Fun one to track down. Had to hop through a couple papers and books, but this is what everything winds up referencing:
> It was an organization designed not only to disseminate information and to clarify issues but also to arouse support for particular symbols and ideas. "The easiest way to inject a propaganda idea into most people's minds," said Davis, "is to let it go in through the medium of an entertainment picture when they do not realize that they are being propagandized."
Koppes, C. and Black, G., 1977. What to Show the World: The Office of War Information and Hollywood, 1942-1945. The Journal of American History, 64(1), p.2, <https://www.jstor.org/stable/1888275>
That paper then points to this ultimate source:
> Elmer Davis to Byron Price, Jan. 27, 1943, Box 3, Records of the Office of War Information, RG 208 (Federal Records Center, Suitland, Md.)
Dead end. I emailed the Archives but the entire facility is closed because of the pandemic. I also tried tracking down the paper's authors but both are retired.
The document likely lives somewhere in this trove: https://catalog.archives.gov/search?q=%22Records%20of%20the%...
Anyway, I'm satisfied it's a real quote (both authors are reputable) though I definitely want the surrounding context now :)