The Staten Island ferry is a pretty good example of this. Unlike the ferrys to Liberty Island (where the statue resides), or Ellis Island (where new immigrants had to go on their passage to the land of the free), this is a regular free ferry service, laid on to serve the guys and galls who live on the Staten Isle. My tax dollars are paying for it, so why not take advantage of the service.
As you can see on the map below, Staten Island is quite a way below Manhattan, much further than Liberty island, etc, but the boat is faster and the trip only takes about half an hour.
The boat trip really is the main draw, as the area of the island around the ferry port is fairly rundown so unless you get a bus deeper in, it isn't really worth a wander.
One attraction that is worth visiting on the island is a huge sculpture of a couple of envelopes, which was built to commemorate 9/11. It's a short walk from the ferry port, and has a nature reserve next to it.
It has the profiles and details of many of the heroes who died serving their country by rescuing people from the twin towers, etc... It's a pretty extraordinary thing.
(So as not to mislead, I should clarify that you pass far closer to the statue of liberty than this, but I just preferred this photo).
Good Statue of Liberty fact, as you probably all know, it was built by the French to commemorate the US independence from their evil british governors. However, you may not know that
a) It was designed and constructed by the very same Gustave Eiffel who had previously built a fairly well known tower in Paris, and
b) while it was designed to be unveiled on Independence day in 1876, the construction overran to such a degree that on Independence day all they had to show was a massive arm and a head. The rest of the statue wasn't completed until ten years later. Relations between the US and France haven't been the same since. Probably.
So to conclude. If you want to get a closer look at the statue of liberty without paying to visit the island, then the Staten Island boat is a pretty damned good way of doing that, but the island itself isn't really worth visiting unless you're prepared to spend the day exploring further away from the dock.