Yes. It is illegal to occupy or to rent out a basement apartment and you have no lease. You are a month to month tenant and the landlord can evict you upon 30 days notice. If you fail to vacate he can bring an action for eviction.
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In New York State (anywhere including in New York City) a tenant without a lease and is not protected by rent control or rent stabilization, is a month to month tenant, and the landlord is allowed to end the tenancy if the landlord serves on the tenant a proper thirty day notice of termination of tenancy.
if you paid rent for the illegal apartment, and your rent period, in the month to month tenancy runs from the first day of the month to the last day of the month, the termination notice must end your "term" on the last day of the month; there cannot be a spill-over into the next rental period. Usually, that means a termination notice that ends the month to month on March 1, 2013, would be legally deficient and a court case for the eviction does not have support and should be dismissed. The correct day to end a tenancy with a notice served on the tenant in January 2013 is February 28, 2013, as long as the notice is served on the tenant at least thirty days before February 28, 2013.
Tenants living in illegal basement apartments cannot be sued by the landlord for not paying rent but can be evicted if the tenancy is terminated by a proper notice. New York law does not provide a simple method for tenants in illegal apartments to recover any rent paid to a landlord.
So check your notice carefully. If the termination notice is defective, you could beat the case and continue to live in the illegal apartment rent free until the landlord comes up with a proper termination notice and the court grants the landlord's request to evict you. For you this may be a big deal. If your notice says March 1, 2013, and the first day of the month is not the last day for your rental period, and your landlord or the judge figures this out (assuming you go to court to defend yourself) the next time the landlord could terminate after March 1 comes and goes is April 30, 2013. So it is important for you to see a lawyer or to learn on your own how to defend a landlord and tenant case.
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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.
Adding to the first two responses, you said "My landlord TOLD me I have until march 1, 2013 to move" - this notice is improper. In NYC, a notice to terminate a month-to-month tenancy must be in writing. See Real Property Law §232-a.