Did the economic downturn of the Depression lead to some sort of moral panic?
I can't really answer the full question but I have a side statement. Assuming you're asking about the US your claim that burlesque clubs were closing down isn't really true- or at least they weren't closing down any faster than any other form of entertainment. What was dying rapidly at the time was vaudeville, and burlesque acts were often taking the place of vaudeville, especially in places like circuses. The theory I've read a few times is that vaudeville was being killed by film as a cheap form of entertainment. However, I know a lot of risque film from the 1920s was destroyed so maybe someone else can elaborate on the cultural shift that inspired their destruction. As far as burlesque goes the 1930s-1950s are considered the golden age, not its decline.
You did see a decline in major spectaculars like the Ziegfeld Follies, but this was largely due to Ziegfeld losing a lot in the stock market and the economy just not supporting major theatrical spectaculars at the time. What replaced them were burlesque stars like Gypsy Rose Lee doing solo acts which were cheaper to produce overall, even when they obtained celebrity status.
Sources:
Online (for dates on Ziegfeld and Gypsy):
I also recently read these for a research paper:
books.google.ca/books?isbn=1845113187
books.google.ca/books?isbn=1550223712
books.google.ca/books?isbn=1134688695
** Edit: I just re-read the wikipedia page on American Burlesque and it also claims that burlesque theatres were being closed down by cities but it only specifically says that NYC mayor Fiorello LaGuardia was cracking down on them. The citation is behind a subscription wall that I can't get through but it comes from an entry in an encyclopedia of pop culture rather than a burlesque or theatre history book/journal so I don't know what the writer's background is. From everything I've read and the burlesque historian (Women's Studies PhD) I've interviewed the 1930s are considered a time of growth for American burlesque, but all my sources for that are in books that I no longer have in front of me so take it as you will.