It is a nightmare that plays out on the streets and subways every few months: A homeless person with a history of mental illness or violence falls through the cracks or wanders away from the system intended to help him, surfaces in a psychotic rage and attacks a random New Yorker.

Though they make up a tiny fraction of crimes, the unpredictable attacks feed perceptions that the city is unsafe and stir demands for action. Politicians send police officers and National Guard members into the subways and pour money into outreach efforts and housing.

And as long-running arguments persist over how to balance public safety and the civil rights of mentally ill people, another outburst inevitably happens.

The debate resurfaced last month when Gov. Kathy Hochul proposed a package of laws to make it easier to take people in psychiatric crisis to a hospital involuntarily, easier to hold them there and harder for hospitals to push them back to the street before they fully stabilize.

But Ms. Hochul’s efforts, unveiled in her latest executive budget, face an uphill battle in the Legislature and opposition from progressives and civil liberties groups.

“Critics will say this criminalizes poverty or homelessness,” Ms. Hochul said last month. “I say that is flat-out wrong.”