As said- you can do the Staten Island Ferry and go right by it for free. It also takes less time. (Just about an hour round trip, if I remember right.)
I do still think it's worthwhile to go TO the island though. I believe it is closed now (not sure if you can go in the museum portion.) They do have a free national parks walking tour of the island though, not sure if it runs in winter. It is a nice tour, though, because you get alot of the history of the statue. They also had a (free) children's craft when we were there in May. (My daughter made a embossed metal picture of the statue.) Again, not sure if that is a summer thing.
The ferry also goes to Ellis Island, which has a nice museum with a movie explaining the immigration process. I'd recommend for kids maybe 8 and up?
The downside is SoL and Ellis Island and the ferry itself could eat up 1/2 the day. I think it's worth it, but depends on what else you want to see.
If you have the option to be in NJ and don't necessarily need to be in all the way to the tip of Manhattan- there are also ferries to SoL and Ellis Island from New Jersey. You can drive right there, plenty of parking. It's the same ferry company as from Manhattan.
I don't know that you need to this time of year, but you also can buy your ferry tickets ahead of time online.