You can try your luck at seeing the Statue of Liberty for free, on the Staten Island Ferry. The Staten Island Ferry is a free ferry service that runs between Lower Manhattan at Staten Island. The route can vary, with some routes sailing closer to the State of Liberty than others. It's worth a shot! You can hop on board at the St. George or Whitehall terminals for a scenic trip across the harbor. Watch out for scammers who may try and sell you tickets - it is free!
90% of people on the ferry will disembark at State Island, turn around, and get right back onto a ferry going back to Manhattan. I'd advise you do the same - we spent an hour or so wandering around Staten Island, but I'd recommend hopping right back on the ferry instead, and exploring more of Manhattan or Brooklyn.