I recently completed a four-night stay at the IHG Candlewood Suites/Harborside in Jersey City, NJ. The property is an eight-twelve minute walk to two PATH (Port Authority Trans-Hudson) rail stations: Exchange Place and Pavonia/Newport. And the hotel provides a free shuttle to and from both stations, as well as the Newport Center shopping mall. Exchange Place is a four-minute ride to the World Trade Center in Lower Manhattan; Pavonia/Newport is about a 15-minute ride to the Empire State Building in Midtown Manhattan.
Cash rates at this property in December might be above the OP's budget, especially after taxes are figured in. But there is a work-around:
Award nights at this property cost 30,000 IHG points/night. When IHG runs one of its periodic 100%-bonus sales on points, you can buy points for 0.5 cents/point, so 30,000 points would cost just $150 USD. And there are no taxes when you redeem IHG points, so you would pay just $150 night.
There are certainly advantages to staying in Manhattan if one can find a decent property within one's price range, and I would recommend that the OP do so if he can. But if he cannot, the Candlewood Suites/Harborside would be a decent alternative.