The good news is that during the summer many students are gone, and now is the time to look for apartments and now is the time to rent so that you know that you will have a place to stay when school starts.
As Danielle said I think that the best way to find apartments near San Jose State University is to walk around the neighborhood and see what buildings you like and what locations you like.
I recommend that you look for apartments as close to campus as possible so that you can get to class without driving a car. Proximity to campus will become very important during the rainy season.
I would recommend walking around near campus as soon as possible. There are usually many for rent signs in the area before school starts. Most are converted victorians or homes in the area but some are apartments. I have not seen many of the apartments near San Jose State on Craigslist but that website could be a good option for you as well. With all the foot traffic traffic near SJSU they usually find renters fast.
Good luck and go Spartans!