New York taxes nonresidents on any income derived from New York state sources.
If I am understanding your questions correctly, your unemployment was from a business within New York. If this is correct, you will need to file a Nonresident Income Tax return for New York.
New York Nonresident filing information
In regards to your next question, if some of the unemployment is from sources outside of New York, then you are correct. You will only report the amount applicable to New York that is from New York sources on your New York Nonresident return.
You will need to report the full amount of unemployment compensation on your home state tax return, but you will be able to claim a credit for any taxes paid to New York on any income taxed twice by both states. This will ensure you only pay taxes one time on that particular income.
When completing your return in TurboTax, be sure to complete the Nonresident New York tax return first in order to ensure the credit for taxes paid to another state, pull into your resident state tax return correctly.
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