Of course it will depend on what kind of car you are building, a restored original or a Hot Rod/Clone, etc. If it's a rarer model I'd keep it as original as possible. IMO the act of painting the bottom of the car to be all new and shiney "Jumped the Shark" a long time ago. A more factory correct look is the current trend that I see more and more these days. To me it looks even more attractive to attempt making them look as they really did. In any event I'd choose between painted and the over sprayed look of my car above, I wouldn't use any kind of Lizzard Skin or other undercoating product, it gives the look of someone trying to hide rust or other ills to most onlookers.

If you are worried about corrosion think about how you plan to drive the car. Most of these cars never see more water than the end of a hose in your driveway to wash them off occasionally, it's not likely the bottom will rot out of it if you don't coat it in some kind of bead liner/POR15/Rust shield product, especially if you do a good job of getting rid of all the rust that is there in the first place and do a proper metal prep before adding a good primer/sealer and quality paint job.