When you enter the W2G into TurboTax you will enter the state information on there. Also, at the beginning when you enter the personal information about yourself, the system asks if you earned any money in a state besides your home state. You need to answer 'yes' to that question.
The system will generate a New York non-resident return for you. New York returns use your entire income from your federal return to figure out your tax rate. Then they charge you that tax rate on the earnings in New York which in your case is just the W2G. So don't be freaked out that your total federal income is on the New York return. They aren't taxing everything, just using it for the math.
@721alucio