Dear Brooklyn Home Owner:
Is the basement an illegal apartment? You really need to check this out yourself. Find the certificate of occupancy for the house. If the house is a single family house on the C of O, then the basement apartment is not legal or safe for that matter, and a landlord of an illegal dwelling is not entitled to use the court system to collect unpaid rent.
If you are giving up on the person illegally renting a room from your mother in her apartment, end the month to month tenancy. In NYC the termination notice is written and served in the same manner as a lawsuit in Housing Court. And, likely, because your mother waited until no time was left to serve a proper notice this month to end the tenancy on August 31, the notice when served in August cannot end the monthly tenancy before September 31, 2014.
The police were involved likely because your mother committed some act that could be viewed as violating a NYC local law (this is a criminal act) prohibiting conduct to evict a tenant without a court proceeding.
A tenant of an illegal apartment (or a room in an illegal apartment) has no legal obligation to pay rent.
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.
Your mother needs to consult with an attorney whose practice focuses on landlord/tenant law to resolve this matter. The process and procedure for a regular eviction is extremely difficult on its own, without the added complexity of the legality of the basement, etc.
Any answer provided by me does not create an attorney client relationship with the author of the question. An attorney client relationship can be formed with me only after a consultation and subsequent execution of a retainer agreement or letter of engagement. Notwithstanding that, the prospective client is afforded the protections of the attorney-client privilege before and during the consultation regardless of whether the retainer agreement or letter of engagement is subsequently signed.