Well, at that time I had a BA in History but I guess that I was choose because I had quite a bit research experience. Nevertheless, the UN is huge and you could be selected regardless of your degree. Deadline was in September-October. But, do they still offer internships? I was an intern there two years ago and I think I was the last cohort. Now, you could try young professional program (YPP). It is a program for entry level positions. Good luck!