The Museum of Modern Art offers several types of internships, from 12-week fall, summer and spring to 12 month internships. Some of the longer ones are paid, as is a specialist publishing internship in the spring.
Even better, MoMA offers sponsorship for J-1 and internship visas for international students/graduates. Find full details on its internships page
For more arts-related internships in the States, check out the International Arts & Artists internship pages