Ameed Ademolu, a close-cropped and bespectacled nurse, descends the street-level stairs to the subway, clad in an orange and yellow vest and carrying a clipboard, paper and pencil. He halts, reaches into a pocket to unfold a K-95 mask and fits it over his mouth and nose. “The mask is for me,” he says, because “I’ve had people spit in my face.”

At the Fulton Street station in Downtown Manhattan, Mr. Ademolu encounters two men, one in his 20s and one older, the latter dependent on a cart piled with his belongings as a walker. The younger one is coherent but agitated, ripping up newspaper and scattering the pieces as he puffs on his vape. Both are homeless and have become friends after months on the subways.

Mr. Ademolu talks to them for half an hour. The older man declines Mr. Ademolu’s offer to have a shelter van pick him up, saying he knows that the shelter system cannot easily accommodate someone who cannot walk up and down stairs. The younger man decides he will go to a shelter. Mr. Ademolu takes the man upstairs to await the van.

Mr. Ademolu spends his days like this, pacing Manhattan subway platforms and stations, searching for people who appear severely mentally ill. He is a member of a Subway Co-Response Outreach, or SCOUT, team, a new project between the city and the state-run Metropolitan Transportation Authority.

His work represents the newer of two strategies that New York’s city and state governments are using to dispel the atmosphere of disorder and danger that permeates the subways. The other is more old-fashioned policing and prosecution. Success for both depends on New York creating a functional mental health system and recreating a functional criminal justice system. If this strategy works, it could become a model for other American cities struggling with crime and mental illness.