Sites and institutions in NYC that are managed by the National Park Service or are part of the Smithsonian are closed, although some national parks remain open to the public without services such as trash collection or public restrooms. 

Closed:

Cooper Hewitt
The National Museum of the American Indian
Hamilton Grange
Federal Hall
Theodore Roosevelt Birthplace
General Grant National Memorial
African Burial Ground National Monument

Open:

National monuments the Statue of Liberty, Ellis Island, and Castle Clinton all remain open to visitors, thanks to funding provided by the state of New York.

Gateway National Recreation Area remains accessible to the public, but without visitor services, such as restrooms or trash collection. This massive national park system in NYC includes Jamaica Bay Wildlife Refuge, Fort Tilden, and Riis Park in Queens, Floyd Bennett Field and Canarsie Pier in Brooklyn, and Great Kills Park, Miller Field, and Fort Wadsworth in Staten Island.

Of course, NYC's many non-Smithsonian museums remain open, as well as parks run by the New York City Parks Department, such as Central Park, Prospect Park, and public playgrounds. 

According to Mayor Bill de Blasio’s office, about 50,000 federal workers in NYC are going without pay, and another 1.6 million more New Yorkers are in danger of losing vital services such as food stamps, school lunch, and housing subsidies. For NYC families dealing with the shutdown, these articles about discounts and resources available may be of some help.

Top photo via StockSnap.io