Big Bad Joburg isn't as bad you heard and common sense should keep you safe. Braamfontein is generally a fairly safe area. During the day you are highly unlikely to experience any trouble in the vicinity. That said, unless you have loads of swagger, avoid walking alone late at night unless in the area close to the hotel. You will see many people walking alone at night in the vicinity of the hotel, as the immediate area has its own security.
Areas to be especially careful and best avoided at night if walking are: Hillbrow; Berea; the CBD across the railway tracks; all the bridges across the railway tracks (including Mandela Bridge); the area near Park Station; Enoch Sontonga Road alongside the graveyard.