You're not going to be able to afford rent on an internship --- most pay very little. Do you have any supplemental $$$ from any source? All of Manhattan is very expensive, not just the Upper East Side!
Keep in mind, you're going to need money for your MetroCard to get around --- $112/month. Calculate what your food costs are going to be as well as your entertainment costs. Add to that clothing, laundry, and other incidental expenses.
Apartment prices in the outer boroughs vary. You can get an alcove studio in a doorman building for $1400/mo. You can also get a regular studio in someone's private home for $950-1000/mo.
If you plan to live in an outer borough, remember to keep in mind how far the apartment is from transportation as well as the time it takes to get into the area of Manhattan where your job is located..
If you absolutely want to live in Manhattan, you can rent a room somewhere but be VERY careful of where you rent!! You can also look into a room-mate situation but, honestly, who wants to share an apartment with a bunch of people you don't know?
OH!! Most importantly and before anyone else says it:
Don't think you're coming to Manhattan to live the "Sex and the City" lifestyle or to "live on the set of "Friends" " Sorry.... just HAD to add that!! There are posters here who think anyone coming to live in Manhattan or NYC in general thinks they're going to live like the girls on SATC or the group on Friends