S.J.
I have lived in several areas of boston and the suburbs and based on the $600/month you want to spend you will need to look farther out of the city. West, south and north of the city are all safe but are you working? If so, that could really help narrow your search. If you live in the suburbs and will be commuting to the city living within walking distance of a commuter rail will save you a significant amt of money. IF not, all commuter rail stops in the suburbs usually have parking lots ( the charge is $5-7 /day). There are some towns that aren't as safe ( mattapan, parts of dorchester, parts of brockton, for example) but overall the city is pretty good.
Best of luck with your move.