Housing
Hopefully you weren't planning on apartment hunting in the Upper East Side. As an intern, you won't be able to afford a room in the Plaza Hotel, so it's important that you do your research. New York University and Columbia University both offer their dorm rooms to interns for the summer. Not exactly inexpensive, the lower end of dorm rooms will run you $3,350 for ten weeks, which includes a mandatory meal plan [1].
You might also want to sublease in the lesser known areas of Manhattan. The average price of a one-bedroom apartment in Inwood is $1,650 a month [2]. You're just a subway ride away from all the action, plus these smaller neighborhoods can have hidden gems of their own.
Daily Expenses
It will be tempting to live like you're in an episode of Friends. Once you're living in the big city, you need to start thinking like a local. You can't spend $20 every morning at that trendy breakfast place around the corner from your office. If possible, prepare meals at home. If your housing doesn't have a kitchen, you can at least stock up on granola bars and other dry goods.
Make sure you're spending wisely on your weekend activities. You can get discounted Broadway tickets at TKTS booths at different locations around the city. Also, don't forget about the free stuff. A stroll in Central Park won't cost you a dime.
Pocket Money
Depending on how many hours a week you work, you may have time to pick up an extra gig. Look into becoming a babysitter or a dog walker. And hey-if you can play the ukulele while riding a unicycle, consider street performing. That way, maybe you will be able to treat yourself to that fancy breakfast every once in a while.
While you may not be raking in cash, having an internship in New York City will make you rich in experience. Need some help budgeting for your big adventure? Check out Central Bank's Money Manager to start achieving your financial goals.
Source:
[1] NYU Summer Housing, New York University
[2] Affordable NYC Neighborhoods Where You Can Still Find a 1-Bedroom, Thrillist