New York City officials declared a public-health emergency as authorities elsewhere in the state announced new measures to halt the spread of measles, stepping up their responses after a recent surge in cases.

The city on Tuesday ordered mandatory measles-mumps-rubella vaccinations and fines for noncompliance in certain ZIP Codes in Brooklyn, where a measles outbreak began last fall in the Orthodox Jewish community. The move is the first time the city has imposed mandatory vaccinations.

“We...