It wasn't that secret. Mostly the secret was because of roosevelt being concerned with his public image and fibbing to people about his recovery. It was helped a lot by the fact that the media saw talking about his disability as a big taboo. But that is exactly what it implies, they may have seen talking about it as a taboo, but in order for that to be true, they must have been aware.

Roosevelt would go to great length to make public appearances in ways to avoid exposing his problem, especially when photos are television was involved. But i'd say anyone interested in politics probably knew about it, though not its extent. Everyone knew he was damaged, but only those who knew him knew exactly how bad it was. This was mostly because of the media taboo combined with the fact that, other than the media, there was very few ways to pass information like this around, so even those who had met him personally only had the ability to tell a few people... and once those people started telling people... well we all know how 2nd hand, 3rd hand, and 4th hand knowledge changes and becomes unintersting.

This isn't so much recorded fact or survey or anything like that than inference from popular fiction from the time, but John O'Hara's novel BUtterfield 8, published in 1935 (set sometime between 1929 and 1932) gives a clue.

Gloria Wandrous (girl about town, 18 years old, New York City upper class but far more involved with drinking and her sex life and her depression than anything else) is talking to Eddie Brunner (also young, raised upper class and held that expectation until the stock market crash in 1929 when his family lost all their money and his allowance stopped arriving, so he took a job as a "night man in a hotel which was more of a whore-house":

Governor Roosevelt's mother is sick and he's going to Paris where she is. She's in the hospital. Did you know that he has infantile paralysis? I never knew that till about a month or two ago. It never shows in his pictures, but he's always holding on to a state policeman's arm. Mm. (O'Hara 137-8)

While Eddie is a little closer to the front lines of the Depression than Gloria is, neither is particularly political. They seem pretty average to me: they know what's going on, but it doesn't affect their day-to-day living and they aren't really seeking out any extra information on it. Passive, even, but they know what's up.

O'Hara, John. BUtterfield 8. New York: The Modern Library, 2003. Print.

I happen to be in the middle of Conrad Black's biography of FDR. Both his run for governor of New York and his 1932 campaign for president included carefully staged events, such as sailing excursions, to demonstrate the candidate's recovery and current vigorous good health.