Are you sure you want the cheapest? Austin is generally safe, but there are areas that are much safer than others. I guess that would be the east side, just north of the river. Also, if you go far south of town, that might be pretty cheap. In general, the farther out from the center of town that you go, the cheaper it is, and east is cheaper than west. You might check the Austin-American Statesman online ads to get a better idea.