Do I need a car in melbourne?

I'm from the U.S. and will be living near the CBD, Parkville or Carlton area for about six months for work. I just read a thread suggesting that many train lines in Melbourne are scary because of potential muggings. My family and I would mostly use public transportation to do some shopping and sightseeing. I also occasionally need to take public transportation from the CBD to Clayton where Monash university is. Is public transportation safe? Should I buy a car?