If you want to avoid a series of (escalating) problems, it's important that you pay your rent on time. Every month. First and foremost, the lease you signed is a legally binding contract. That, in and of itself, should be reason enough. However, if for some reason you feel like you need an additional incentive...
There are other obvious reasons you should pay your rent on time: your landlord will like you more than if you don't pay your rent on time, putting yourself in a relatively better position to negotiate come lease renewal time; you won't be charged late fees (remember though: late fees are only allowed IF it's stipulated in your lease) or face eviction; and you'll feel like a good citizen and responsible adult. All indisputably true.
It's also not fair to the landlord if you don't pay on time. He/she needs your rent check to run their business... you are their income. Would you appreciate it if your employer withheld your income a few days, or a week, beyond when you were supposed to be paid? I thought not. Remember, not every landlord is rich, or has lots of extra cash they can depend on if you don't pay your rent on time.
But now add another reason residents of NYC rental apartments should make every effort to pay their rent on time: it now goes on your credit rating.