It can be done. And NYC is pretty magical in the snow.
Keep in mind that with the subway system below the streets, there are very few places where snow sticks and accumulates due to the heat underground. Since public transportation rarely shuts down, and we have a huge sanitation department to clean up, salt the streets and plow, the city does not shut down. Shops and restaurants will be open, business as usual, and most tourist attractions will still operate. Depending on your plans, you can still have a great trip. The only obstacle will be if you need to get to or from an airport for a flight, or if you were planning to stay somewhere other than Manhattan. Outer boroughs and NJ will have snow accumulation, and roads will be worse outside the city.