Yes. You must commence an eviction proceeding. Technically, the tenant does not have to pay rent if basement is illegal. I strongly suggest that you hire an attorney.
Yes you can evict the tenant. However, I strongly urge you to seek the advice and/or assistance of an attorney to discuss your options.
Dear I just want to know?
Perhaps I might suggest an answer.
Your renovated basement may be "code" but may create an issue with the building's existing certificate of occupancy. If the basement renovation was made without approval of an alteration plan from the NYC Department of Buildings, then it is not likely, that the certificate of occupancy, was updated to include the new apartment in the basement.
A landlord without a certificate of occupancy for a dwelling runs the risk that the tenant being evicted will retain an attorney to try to recover all or part of the rent paid for the "illegal" apartment, in particular, when the tenant had no reason to have knowledge that the rented apartment was illegal before being sued. In your case, a court may see more in favor of the the tenant, as he was not a stranger, but a family friend. Whether a court views your taking rent from a family friend for an apartment which does not appear on the certificate of occupancy, may depend on factual proof that may establish that the apartment may be legalized or may not be legalized. In the case where the apartment is capable of being legal, the court will feel less stressed about the failure to obtain an amended Certificate of Occupancy.
I agree with the two attorneys and their earlier answers. You need to bring your legal concerns to an attorney and let your attorney explore with you the choices you may have other than pursuing an eviction case in the Housing Court (By the way, a forgotten form of eviction case still may be used used, which avoids the problems in the Housing Court. That form of eviction is known as an Ejectment and may be started in the regular part of the Civil Court, in certain circumstances.)
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.