Your NY unemployment is NY-source income and thus taxable by NY. As to whether you must file an NY tax return, see "Who Must File" on page 7 of this reference: https://www.tax.ny.gov/pdf/current_forms/it/it203i.pdf
Your NY unemployment is not taxed by NJ. Unlike NY, NJ does not tax unemployment benefits.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
