Congress followed Roosevelt's lead and supplied $350,000 for the expedition - about $6.1 million in today's dollars (private funding also helped, and government agencies were able to assist in other ways, e.g., equipment loans).
A review of the records of the U.S. Antarctic Service (located within Record Group 126 of the National Archives) shows that the expedition's accomplishments included extensive aerial & photographic surveys of the continent's geography; the construction of a meteorological outpost which made twice-daily weather reports; measurements of how high ice shelves were above sea level; analysis of magnetic readings; aerial "cosmic ray observation"; examination of volcanic rock; and the study and gathering of biological samples. With respect to the latter, the U.S. National Museum (the Smithsonian) wrote a letter on July 8, 1941 acknowledging receipt of many specimens, including "103 birds, 3 fishes, a collection of plants (mostly lower cryptogams), 1 octopus and 1 lot of shrimp..." The expedition also attempted to bring back live specimens, for the National Zoo, but I'm not sure how successful they were. For example, they took aboard crab-eater seals for the return voyage but the animals wouldn't eat and became extremely foul-tempered. Most or all of them seem to have died.
In 1942, the Department of Interior reported that "Approximately 2,000 miles of continental coastline was discovered and mapped on aerial and dog-sled thrusts into previously unknown areas..." by the U.S. Antarctic Service (Annual Report of the Department of the Interior, fiscal year 1941, p. 474).
Above: In this footage, put together by Extreme World, we see a series of black & white and color video segments of the Snow Cruiser, including its unloading in Antarctica, which almost ended in catastrophe. This video shows the true size of this enormous vehicle. YouTube link: https://www.youtube.com/watch?v=ABaxjIRxheQ.
It doesn't appear that any of the purely New Deal agencies (WPA, CCC, etc.) played a large role in the 1939-1941 expedition to Antarctica. However, there are several interesting loose connections to the expedition, as well as to antarctic exploration generally.
"Worked during the morning trying to find something to do. There are so many WPA fellows, Navy Yard employes on hand that we fellows of the expedition are having a tough time finding work" (Tim Baughman, Ice: The Antarctic Diary of Charles F. Passel, Lubbock, TX: Texas Tech University Press, 1995, p. 2).
This would seem to indicate that WPA workers help load supplies and equipment onto Passel's ship, the North Star. This could have been a planned project, or it could have been more of a spur-of-the-moment job. Sometimes, WPA workers would be diverted from their regular jobs to assist on other projects, for example, search & rescue, flood control, or firefighting, if there was a pressing need.
It is also possible that the WPA played a role in the lumber and pre-fabricated buildings that were taken on the expedition, but I haven't found conclusive evidence of that yet. The buildings were designed by Major Andre Leonard Violante of the U.S. Army Quartermaster Corps, and constructed by the Putnam Lumber company in Shamrock, Florida (see, e.g., "Byrd's Antarctic Expedition Takes Already-Built Houses," Abilene Reporter-News, November 19, 1939, p. 23). The Quartermaster Corps was one of the biggest employers of WPA workers. For example, during fiscal year 1940 it employed 26,500 of them (Federal Works Agency, Report on Progress of the WPA Program, June 30, 1940, p. 44). We also know that the WPA employed all types of unemployed Americans, including architects. Is it possible that a WPA architect (or other WPA workers in the Quartermaster Corps' architectural division) assisted Major Violante in some way?
In congressional hearings on the 1939-1941 antarctic expedition, the following was noted:
"The sum of $10,000 will be needed to supply a quantity of fur clothing and sleeping bags for the personnel
at each continental base who will go on the trail or participate in airplane flights. This estimate has been discussed with a representative of the Indian Arts and Crafts Section of the Office of Indian Affairs and orders have been placed with this service in Alaska, subject to telegraphic confirmation ("Expedition to the Antarctic Regions, Hearings Before the Subcommittee of the Committee on Appropriations, House of Representatives, Seventy-Sixth Congress," p. 4). Later, in the 1940 annual report of the Department of Interior, it was reported that "native craftsmen sold $7,300 worth of mittens, caps, mukluks, parkas, trousers, and robes to the Byrd Antarctic Expedition" (p. 398).
The "Indian Arts and Crafts Section" or, more appropriately, the Indian Arts and Crafts Board, was created when President Roosevelt signed the Indian Arts and Crafts Act into law on August 27, 1935. This law was part of the overall "Indian New Deal," facilitated by John Collier (Commissioner of Indian Affairs) and the Department of the Interior. The goal was (and still is today) to enhance the market for genuine American Indian arts and crafts.
The WPA assisted in compiling and publishing (and perhaps analyzing too) meteorological data from Admiral Byrd's first two missions to Antarctica (1928-1930 and 1933-1935). The lead author of the 377-page report, George Grimminger, wrote:
"Since, owing to the demands of his other duties, Haines [Grimminger's colleague] found it impossible to devote any time to the work, I then agreed to carry it on alone; after working alone some time, it became apparent that unless assistance could be obtained, the task would be a long and arduous one, and it is indeed fortunate that I was able to obtain assistance provided by the Works Progress Administration, assistance which I gratefully acknowledge and without which the completion of the present volume would have been considerably delayed."
(G. Grimminger and W.C. Haines, "Meteorological Results of the Byrd Antarctic Expeditions, 1928-30, 1933-35: Tables," Monthly Weather Review, United States Department of Agriculture Weather Bureau, Supplement No. 41, October 1939, p. iii, accessed January 9, 2017, emphasis added).
Let's hope that Roosevelt's forward-thinking on science... overcomes today's backward-thinking on science.