The long view: The basic laws of economics aren't going to suddenly change, as long as a web service provides something people need or want then it should have a shot.
The biggest potential change here is that in an economic downturn what people want and need may change drastically. This may be disruptive for existing startups, but it will create a lot of space for new ideas to meet those new needs.
Things that are imminently practical like new and better ways to search for jobs, or ways to trade skills or items for other items could have real practical value in a depression. Sites that allow people to group their buying power and purchase a larger quantity of some item at lower prices, or sites that help you find carpool partners or generally make your personal economics more streamlined should enjoy increased success in a depression.