Southwest will do fine because they didn't falter the last time we had an economic downturn a decade ago. Thanks to Southwest's extremely low seat-mile costs, they may actually thrive as the airline takes over business reliquished by the cutbacks to AA, CO, DL, NW, and UA.
JetBlue is another airline that will do fine. Like Southwest, JetBlue sports low seat-mile costs and has locked up some important markets like JFK-FLL.
In Europe, both EasyJet and Ryanair could become major players in the European market, especially between the British Isles and the European continent. Like Southwest and JetBlue, both these low-cost carriers have low seat-mile costs and will definitely benefit as AF, BA, IB, LH, etc. cut back their services to and from the British Isles. Indeed, with a glut of 737-300/400/500 planes suddenly becoming available, Ryanair's intent on getting more CFM56-powered 737's could end up being less expensive that Ryanair originally envisaged because of the depressed prices for these planes.