It's not helpful to say "it depends" but to give some context, let's say it depends on whether you live alone or with roommates.
You will pay a minimum of ~$800 per bedroom (+utilities) for any place in Manhattan, and that's assuming 3+ total bedrooms as well as the option to live uptown in Harlem or similar. More likely you will pay a minimum $1,200 for your room + utilities.
Add groceries @ $80 /week (that's if you don't buy organic or many name brands), "going out money" @ $150 / week (restaurants, bars, taxis) and miscellaneous expenses like stuff for your home, metro cards, etc.
Now we're around $2k month and that's not a particularly active or comfortable lifestyle. If you want to save on top of this (trust me, you do) then you'll be spending $2,500 /month easily just to scrape by with roommates, a tight 'fun' budget, and minimal savings.
After taxes at a regular job that means you need to make at least $45k salary.
Now, if you live alone:
You will pay a minimum of $1,600 for a studio apartment, and again that requires willingness to live uptown (although not necessarily as far north as Harlem) and most likely in a walk-up or at best a building with an elevator but no doorman (my personal situation is this @ $2,200, 25th/3rd).
Add name-brand groceries @ $100+ /week and more going out + miscellaneous funds + more savings (the bigger they are, the harder they fall).
We're now around $3,500 /month in total expenses (including ~10% income savings), translating roughly into $60k salary.
This is an OK lifestyle, but given most building income requirements (annual income >= 40x monthly rent), you would not meet the $1,600 * 40 threshold and will need either more income (freelance, parents, student loans, etc) or a guarantor.
In addition to all this you will need furniture, first/last deposit + security, potentially broker fees ($3k+, one-time), and other "moving to NYC" things.
So you now need savings of $8k+ in addition to a healthy salary ($60k+), which is effectively 3 months' salary from either another job prior to NYC or a higher salary ($80k+) while already in NYC.
Basically, this place sucks and everyone is secretly dependent on their parents.