The Staten Island Ferry is the best free activity in New York. The 25-minute crossing from Whitehall Terminal in lower Manhattan to St. George on Staten Island passes the Statue of Liberty, the Ellis Island immigration station, Governor's Island, and the full Lower Manhattan skyline — some of the most recognisable urban scenery in the world. The ferry runs 24 hours a day, every 30 minutes during off-peak hours, every 15 minutes at rush hours. It costs nothing.
You don't have to get off at Staten Island (though you should at least look around — Staten Island is the most overlooked borough, with a good art museum and the largest remaining fragment of primeval forest in New York City). Just ride across and ride back. The views from the upper deck at dusk or dawn are among the most spectacular available in any city, and they're completely free.