With vacancy rates around 1% in many NYC neighborhoods, most landlords don’t feel that they need to negotiate. Exceptions include an apartment that has been on the market for a long time, one that has a lot of similar competition in the same building, one that is particularly undesirable, or is in a building with a history of bed bug problems.
Seasonally speaking, you may encounter more flexibility in the slowest part of the rental cycle, from November to March. Offer to move in immediately and let the landlord know you are willing to sign a lease of 15 or 16 months instead of a year, enabling them to re-list the apartment during the busy summer months.
Another tactic to try is to negotiate on amenities. For example, see if the landlord will waive the gym fee or bike storage fee, or pay for a minor improvement to the apartment like replacing an appliance, refinishing the floor, or soundproofing the windows.
In the slower winter months and when a brand new building is leasing many apartments at once, landlords may offer rental concessions--usually in the form of paying the broker's fee and/or kicking in one or two month's free rent. (If they're not offering them, be sure to ask.) The rental credit usually only applies to the last month of the lease, which will be free, while the apartment will be advertised with a "net effective rent." This is not the amount you'll actually pay per month -- that is the "gross'" rent. So if your monthly rent is $3,000, and you get free rent at the final month of the lease, your "net effective" rent is $2,750. Moreover, if you renew your lease, bear in mind that the increase will be based on the gross rent, not the net effective rent.
If you break the lease, many landlords will revoke the free month.