Walk around or stay in place? Both. "Street photography" can mean different things to different people, so my reply is in the context of unposed photos with people in them, rather than posed portraits or urban architecture.
Stay in place: This is what I do when I find a good scene due to lighting or the setting. I'll think about the shot I want, compose, keep the camera away from my face, then snap as people walk through the scene. The 1st photo below is an example of waiting.
Walk around: I see much more when I walk. That gives me more opportunities for inspiration, I see more architecture and other things as I move, and I can do some photography while commuting. My technique for shooting while walking is wide angle lens, manually focus around 8 feet in front of me before I start walking, stop down to get extra depth of field, and pick a very fast shutter speed. TAv mode works well for that. See 2nd photo for example.