Taxes on an NY summer internship

That's just over 12K. You will probably get around 1k taken out for the whole year.

Ask your employer for the State tax income form and there is an option to add an additional amount on to take out for state taxes. I used to have the state take an additional $5 out for taxes each check just to be safe.

Also claim 0 for dependents

The good news is most likely you will end up getting around $300 back when you claim your taxes.