People give away their labor in this case because it makes them feel good, and/or because of the non-monetary community rewards surrounding the effort. Neither of those changes materially as the economy goes down (or up).
Actually, a downturn in the economy would result in an increase in unemployment which would probably result in more people with time on their hands so more people doing volunteer work. It would also result in big corporations having less money to spend. So, a downturn in the economy is *good* for Wikipedia and the rest of Web 2.0.