The Greek Pericles, a statesman and defender of democracy during the Peloponnesian War, stated that democracy by its very definition doomed to failure. For, as a government of the people, how can each man's "belly" be satisfied without neglecting or harming someone else's, Pericles asked.
There were enormous, often violent strikes and protests in New York City, Chicago, and elsewhere. The Communist Party, as #3 mentions, had hundreds of thousands of members, with many intellectuals idealizing the "experiment" being attempted by Josef Stalin in the USSR. Lynching in the South reached a peak during this period, right wing organizations, including the Nazi Party itself, attracted great interest. Veterans demanding payment of pensions were violently evicted from Washington DC. In 1934, textile workers throughout the South, some with radical ties themselves, went on a massive strike, and some local governments used violence to try to break them. It was a very turbulent time, and this extreme social unrest provides a bit of important context for the New Deal. There were no shortage of demagogues on the left and right, but most did not reach a national audience.
Democracy was probably as fragile in America in the 1930s (and elsewhere around the world) as it has ever been. The Great Depression, of course, had an enormous amount to do with this. Radical political movements of both the right and the left were probably more popular then than they have been at any other time in American history. I'm guessing, for instance, that the 1930s were the heyday of the American Communist Party, simply in terms of the numbers of members and the prestige of the people involved.