It's only fair that I discuss housing options in New York City as well as Los Angeles. I interned in New York City the summer after my freshman year of college. I lived at the NYU Dorms (Hayden Hall) through the NYU Summer Programs. NYU and a few other schools in the city open their doors to students from schools all over the world for the summer. Interns populate most of these dorms over the summer.
Below are some great resources for internship housing in New York City:
NYU Summer Program: http://www.nyu.edu/summer/
Perk: I was able to use the gym at NYU and get on the NYU student meal plan for the summer !
Educational Housing Services: http://www.studenthousing.org/what-we-offer/interns
The New School Housing: http://www.newschool.edu/studentservices/housing/suinterships.aspx?s=11:5