May, September, and October are the best months for Princeton. The campus is at its most photogenic in late spring (flowering dogwoods and tulips in the quadrangles) and in mid-autumn (the maples around Nassau Hall and the chapel turn fierce red). The weather is dependable, students are around, and the town has its proper rhythm.
Reunions at the end of May — when 25,000 alumni descend on town for a long weekend — is a unique experience if you happen to be in the area, but a difficult time to find a hotel. Commencement the week after is similar. August is hot, humid, and the campus is empty; January and February are cold but the campus in snow is genuinely beautiful and rooms are easy to come by.
The town empties dramatically during the winter and summer breaks; if you want the place at its liveliest, time a visit to the academic year between mid-September and early May, ideally on a weekday.