Non-residents of New York are subject to NY income tax on NY-source income. This web reference details what is considered to be NY-source income:
https://www.tax.ny.gov/pubs_and_bulls/tg_bulletins/pit/ny_source_income_nonresident.htm
Your home state of CA can tax ALL your income, regardless of its source.
If you are subject to NY income tax, you must file a non-resident NY tax return in addition to your home state CA tax return. CA will give you a credit for the taxes paid to NY on any income that is taxed by both states, so in effect you won't be double-taxed. With TurboTax, you should complete your non-resident tax return before your home state tax return. The program will then calculate and apply your credit.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
