Recessions can be a mixed bag for entertainment conglomerates. On the plus side, cash-strapped consumers often favor “staycations” over exotic trips — and thus are more likely to go to the movies or watch TV. But recessionary downsides abound too: Those same consumers usually ride herd on discretionary spending, which means they buy fewer DVDs and video games. And woe to the companies overly reliant on advertisers.

Philippe Dauman, CEO of New York–based entertainment giant Viacom, isn’t mincing words about the impact of the current downturn. “Overall the recession’s been a drag,” says Dauman, a Viacom veteran who left in 1999 to start his own private equity firm but returned as chief executive in 2006.

To be sure, plenty can go wrong in a sprawling media empire. Despite being blockbusters, movies like Star Trek and Transformers: Revenge of the Fallen...