In the great scheme of things, which airport you fly into is largely irrelevant. Kennedy would probably be the last choice of the three if you're staying downtown, but even that is no big deal. There's a brand new light rail link from the Kennedy terminals to the subway if you're brave. As with most things like this, you should shop price and convenience.
If you're only spending a day in New York you should protect your investment by staying in one of the better NYC hotels, between 6th Avenue (Avenue of the Americas) and 8th Avenue, and between 42nd and 55th Streets. This gives you the best access to most of the things you'll want to see on a first visit, is convenient to all major subway lines, and will minimize wasting time on travel within the city. Oh, and you should plan on getting to the ship relatively late on the day of sailing. You'll be on the ship for a week or more, so there's no sense sitting on it all day on your day of departure when you could be enjoying the one of the great cities of the world. If you're nervous about check-in, do that as early as possible, dump your bags, get your cards, then get back outside and see some more of NY.
Despite all the jokes and dire advice, use the subways if you can. They've been pretty well cleaned up, and they're usually the fastest way to get anyplace. Subway maps are readily available. Don't be afraid.