With "some fact-based suspicion," the police can stop anybody, interrogate him, and search him. (Or her.)
But the police seem to have trouble distinguishing between suspicious appearance and conduct. The appearance route seems to result in mostly Black and Brown people getting stopped and frisked, because just having dark skin makes a person suspicious.
If you protest, you might get arrested. That's what happened to Leonardo Blair, a reporter who was stopped and frisked and then arrested and charged with "making unreasonable noise" and "disobeying a lawful order." The charges have since been dismissed.