There is no cheap housing anywhere in LA. But your best bet is to try Burbank for a good mix of safety and cost. It's just a bit east of Studio City.
You might also see if there are any apartments in your budget around Vineland Ave and Ventura Bl (Studio City), like on Arch Dr, Riverton Ave, Bluffside Dr, and a bit north on Vineland Ave, like around Aqua Vista St and Acama St.
Have you asked the school where they suggest you look for housing? Even if they don't have on-campus housing, most schools have off-campus housing / roommate suggestions.