I'm reading Nancy K. Bristow's book American Pandemic: The Lost Worlds of the 1918 Influenza Epidemic, and she mentions a US Public Health Service poster drawn by a "well-known Washington cartoonist.

The poster is titled "Use the handkerchief and do your bit to protect me!" The National Library of Medicine has a scan online.They don't credit the artist though.

It was pretty obvious that Clifford  Berryman is the cartoonist. A little more research found the article about the poster that Nancy cited, Droplet Infection Explained in Pictures (Public Health Reports 33: 46, November 15, 1918) is online in the Internet Archive.