Brokers provide a service like any other service. You dont have to use one, just like you don't need an attorney for everything attorneys are hired to do.
If there is a particular building you are interested in all you need to do is find out who the management is. Legally, all residential buildings in NYC have to have a sign with the contact info of the landlord/management in the lobby. Just get into the building and call the number you find to inquire after any vacancies.