In my experience, saying "Gut Shabbos" to a member of the opposite sex who you don’t know too often leads to the grossest forms of immorality. As I walk down Pico Blvd on a Shabbos, I only say hello to Jewish men. I completely ignore women unless they’re really hot and I’m willing to risk my soul for them (or unless they’re shiksas or friends of mine).
The custom in Pico-Robertson Orthodox life is not to greet Jewish members of the opposite sex when walking down the street.
One thing I can’t figure out is when you see someone in the distance walking towards you. Do you stare at them until you pass (that can be 20 or 40 seconds, a definite violation of my parole terms) or do you do as I do and look off to the side and pretend to be absorbed in a tree or a billboard until they’re right up on you and then turn and greet?