Apartments near UTSA are in every direction. There are a good number of student focused apartment communities within walking distance to campus on each side, with the majority directly North or directly South (~15 apartments within 2 miles offering rent by the room). If not within walking distance there is a UTSA shuttle which also serves off campus apartment communities (http://www.utsa.edu/auxiliary/shuttle.html).
With San Antonio being a very large city there are many apartment communities within commuting distance. That said, most students live within two miles of UTSA to avoid the hassles of driving to campus and because you can find the full spectrum of apartments, from budget to luxury.