In the old days, my work used to take me to NYC for 1 week at a time every six to eight weeks so I came to know it pretty well. The trouble is, it's a dynamic place and changes quickly so anything I can say may well be out of date.
First off, don't get too paranoid about violence and street crime. In 15 years of visiting every six weeks, I never once had a problem (except for being in a taxi that decided to have an accident). Do take normal precautions but don't let it spoil your trip.
Yes to the Empire State Building and I agree that night is a good time to see it...the lights go on forever. It costs money now but the next time your wife makes you watch Sleepless in Seattle you can say "I was there".
Personally, I wouldn't bother landing and going up the Statue of Liberty. It's best seen from a distance and is a bit boring inside. My advice is to avoid the expensive tourist boat and just jump on the (free) Staten Island Ferry. It goes right by the statue and gets you some great photos. There's also some gorgeous skylines from the boat, particularly if you go so you're coming back around dusk.
For another classic view, grab a taxi to the River Cafe just across the Brooklyn Bridge from Manhattan. Don't eat there--overpriced and so so food--but have a drink or two and enjoy another view that turns up in half the films you see about NYC.
Eat lunch at a deli. They're all good. Have corned beef on rye with a dill pickle and marvel at the amount of meat and size of the pickle. Deli cheesecake tends to be good too.
If you want to see a Broadway show and are willing to take your chances, line up at the Tkts booth in Times Square for cheap (well, less expensive) tickets for a same-day show. There's a TKTS web site that explains how it works.
I tried the "stay in New Jersey" thing last time I was there (I stayed with friends there) and twice in one trip a broken down train blocked the tunnel and caused many hours of delays. Just sayin'.
For hotels, MAYBE look at the Empire on Broadway near the Lincoln Center. It used to be okay but not quite as expensive as others in the same area...however I have to qualify this by saying I know it's been renovated since my last stay.
If you want to blow the budget on a "trip of a lifetime", the Ritz Carlton on Central Park Sound and Le Parker Meridien on W56th Street are both incredible but sit down before checking the prices. (Travelling on expenses was fun....)
I'll see if I can think of anything else.
Edited to Add: Yup. I thought of something else. Make sure you have good travel insurance that's valid in the USA for medical. The aforementioned taxi accident would have been a big problem without insurance...and the experience of being in Emergency with a bleeding head wound that nobody was treating until they knew they'd be paid was kinda sobering (and made me love the NHS even more).
and check out some of freaks or the peep shows.
