South Berkeley IS a large area, but it does NOT include the Elmwood, which is actually an Eastern part of Berkeley. Southeastern, but no one in Elmwood considers it South Berkeley and no one in South Berkeley think of themselves in the Elmwood. South Berkeley is technically that part of Berkeley south of University and West of Shattuck. I lived and owned in South Berkeley for almost 20 years, in various locations, and I love it. There are good and less good areas, and a very few areas that some might consider rough. Everything about 4 blocks south of Dwight is considered highly transitional, with some great streets and some that are more affected by various kinds of crime. Some, like the area closer to the new Berkeley Bowl, which is South west Berkeley, but very much defined as SB, are transitioning or gentrifying faster than others, where there are, say, higher density apartment buildings.
"Safe" is a highly interpretive term and it largely depends on personal threshhold. There is not a neighborhood anywhere that does not have car thefts and burglaries. My personal definition of safe means higher frequency of crimes with a component of violence....so armed robbery, or worse, and known houses where unlawful activity has been documented..
It is highly advised that for ANY property, in any neighborhood, if safety is a concern for you, that you contact the community liason officer for the area to see what types of crime have been reported for the immediate area, whether there are specific households that are known for criminal activity, and whether the neighborhood is organized for crimewatch and emergency preparedness. The CLO can give you the contact info for the crime watch group, and that person can usually give you a real sense for which direction the neighborhood is moving and how organized the neighbors are.. Meeting neighbors who are actively working to improve the community will make you feel much more comfortable with the street you are moving in.
Remember, even in the so called "best" neighborhoods there is crime and the occasional problem neighbor, If you do your homework, any neighborhood can be a good move for the right house at the right price.