The extent to which Roosevelt hid his disability is actually debatable. There's absolutely evidence to suggest that the White House was hostile to depictions of Roosevelt in his wheelchair, (even going to far as to sic Secret Service agents on photographers).^1 That didn't seem to completely stop the press, however, as fairly detailed descriptions of his disability were released throughout his political career. Both TIME and Life Magazines ran articles describing his "lameness" and his extensive use of a wheelchair.^2 Therefore, it seems like a half-measure. The White House resisted the release of images of the President in his condition, but it doesn't seem it was any secret that he was a polio survivor.

The question is, therefore, how they spun it! Roosevelt convinced many within his personal circle that he was, in fact, getting better and regaining his ability to walk. His famous "Two-Point" walk was used to make it very short distances, (from a chair to a podium, from a car to a building, etc).^3 Roosevelt referenced his physical ailment for the first time in 20 years at a congressional address, when he apologized for sitting down.^4

So to summarize, it was absolutely not some state secret that the President was polio-stricken. The extent of this disability was, however, sensitive. Public perception was likely mixed. Some would have read articles describing his use of the wheelchair, fewer would have seen rare photographs of him actually in it. I'm inclined to say people were generally well-aware of his disability, but likely saw it as an ailment he had overcome, rather than the daily struggle it was.

Sources:

1: Goldberg, Richard T. (1981). The Making of Franklin D. Roosevelt: Triumph Over Disability. Cambridge, Massachusetts: Abt Books. ISBN 9780890115640.

2: http://ideas.time.com/2013/07/12/the-myth-of-fdrs-secret-disability/

3: Ward, Geoffrey C.; Burns, Ken (2014). The Roosevelts: An Intimate History. New York: Alfred A. Knopf. ISBN 9780307700230.

4: "Address to Congress on Yalta (March 1, 1945)". Mar 1, 1945. Retrieved Aug 12, 2013.