It is highly recommended that graduate students arrive early in the Fall (August) to secure housing. New graduate students have a number of options for housing. One inexpensive option is to live in UCSD Associated Residential Community Housing (ARCH); here you'll find four different apartment complexes near the campus, owned by the university and expressly devoted to housing graduate students. As these are in high demand, there is a waitlist to get in. You can put your name on this waitlist as soon as you are accepted to UCSD. Note that it often takes one year or more to get an apartment off the waitlist, depending on your preferences. A faster option may be to apply to be a roommate of someone already living in university-affiliated housing; these forms are also available on the website.
Another option is to rent a non-university-affiliated apartment or house in the area. There is a wide variety of price ranges and neighborhoods, and many are serviced by free UCSD shuttles or subsidized San Diego city buses. For example, the Superloop city bus routes run frequently to and from UTC/La Jolla, the apartment-filled area of town directly to the south of campus. Other popular (but farther from campus) neighborhoods include Clairemont Mesa, Pacific Beach, Ocean Beach, Hillcrest, North Park, and University Heights. You can find both rental and roommate listings at the UCSD Off-Campus Housing Directory provided by the UCSD Commuter Resources.