New York reported its first two coronavirus deaths Saturday, as the state faced a growing crisis that has left hospitals worried about a lack of beds, partially closed the State Capitol, put dozens of city firefighters in quarantine, raised pressure to shut schools and prompted the cancellation of Catholic church services.

As of Saturday evening, the state had 613 cases of coronavirus, with more than 100 people hospitalized, state officials said. At least 183 of those cases are in the five boroughs, city officials said.

Gov. Andrew Cuomo said an 82-year-old Brooklyn woman, who had emphysema, died early Saturday morning. She was admitted to the Wyckoff Medical Center in Brooklyn on March 3 as one of the city’s first coronavirus cases.

The medical examiner in Rockland County, north of the city, said Saturday a 64-year-old who died March 12 in the Village of Suffern had tested positive for the virus and “had other significant health problems which were likely contributory to death.” A spokesman declined to release any additional details.

“We’ve known from the outset that these people are the most at risk in this pandemic, and today’s news is a sad confirmation of that reality,” New York City Mayor Bill de Blasio said.

Mr. Cuomo said he has increasing concerns about the state’s hospital capacity. There are roughly 50,000 hospital beds statewide, with 3,000 intensive-care rooms. Those rooms would be needed for the most severely ill people, who have underlying health conditions.

On Saturday morning, NewYork-Presbyterian, one of the largest hospital systems in the New York City area, said that it would be stopping all elective surgeries and procedures beginning March 16. The move frees up the system to accept more coronavirus patients. Other hospitals are expected to do this as the virus spreads.

“Nobody is really talking about reducing the ultimate spread,” of coronavirus, said Mr. Cuomo, “but reducing the rate of the spread.”

Mr. de Blasio also said that a city firefighter had tested positive for the virus. The member and 33 other firefighters will be self-quarantined and the firehouse will be decontaminated, the spokesman said. There are currently 99 fire department members under self-quarantine, according to the FDNY.

Throughout the state, churches, prisons and schools are taking steps to fight the virus.

The Roman Catholic Archdiocese of New York said Saturday that it was canceling Mass at churches in Manhattan, the Bronx, Staten Island and Hudson Valley counties. Churches will remain open for private prayer, the Archdiocese said in a statement.

The New York State Capitol will close to visitors starting Sunday after two members of the state Assembly tested positive for the virus, Mr. Cuomo and Assembly Speaker Carl Heastie announced Saturday.

Assembly members Helene Weinstein and Charles Barron, both Democrats from Brooklyn, haven’t been in Albany since early March, Mr. Heastie said. Hundreds of lawmakers, their aides, lobbyists and activists work in and visit the Capitol on any given day when the state Legislature is in session.

New York City Council Speaker Corey Johnson told members of the body that its offices in City Hall and a nearby building would close starting Monday. The Council is suspending requirements to hold its regular session as well as land use and budget hearings, Mr. Johnson wrote in a letter.

New York Gov. Andrew Cuomo visits the state's first drive-through Covid-19 mobile testing center in New Rochelle. Photo: Marcus Santos/Zuma Press

And the agency that supervises the state’s prison system said it was suspending in-person visitations to inmates starting Saturday evening. Legal consultations would still be permitted, and inmates would get five free stamps and one free phone call a week to stay in touch with family members.

A student at I.S. 27 on Staten Island has also tested positive, the mayor said, but wasn’t at school last week. The school is being cleaned this weekend and is set to open on Monday, Mr. de Blasio said.

Speaking at a Saturday press conference, Mr. de Blasio said he was still resistant to shutting the city schools. He said he wasn’t yet prepared to impose further restrictions on New Yorkers to the level of citizens in Italy, where only pharmacies and grocery stores are open and residents are all confined to their homes.

“We want to strike a balance and we’re trying to discern that balance every day,” Mr. de Blasio said.

Across the Hudson River in Hoboken, N.J., Mayor Ravi Bhalla announced a citywide curfew starting Monday. Residents were ordered to stay in their homes between 10 p.m. and 5 a.m. except for an emergency, or if required by an employer.

Meanwhile, Mr. Cuomo said that on Friday 150 vehicles came through the state’s new drive-through coronavirus testing site in New Rochelle, N.Y., a New York City suburb that is the epicenter of the state’s outbreak.

Each vehicle took about 15 minutes to process, which means that more people can be tested faster, he said.

The state is planning to soon open a drive-through coronavirus testing site in Long Island. Mr. Cuomo said the state was considering the parking lots at Jones Beach State Park as a possible location.

Corrections & Amplifications

A 64-year-old who died March 12 in the Village of Suffern had tested positive for the virus. An earlier version of this article incorrectly said the person was 65. (March 14, 2020)

Write to Melanie Grayce West at melanie.west@wsj.com and Jimmy Vielkind at Jimmy.Vielkind@wsj.com