South Yarra is a safe suburb, but like most areas in Melbourne (and the world) it's not a great idea to be walking around by yourself in the middle of the night. Don't let that put you off moving to South Yarra though as it is a very safe suburb.
I have been skipping through the streets of South Yarra for over 30 years, and have raised (and schooled) children in the area. I have never had a time where I have been frightened, or felt threatened to be alone. My home has never been broken in too. Its a busy area, always with people around, there for I feel quite confident to say that South Yarra is very safe.