There are several areas of Manchester that have safe neighborhoods. The mall is towards the south side of the city. There are not many neighborhoods surrounding the mall. You can visit the city of Manchester website (
Welcome to the City of Manchester, New Hampshire - Where History Invites Opportunity! ) to gain access to the statistics such as crime rates. You can base neighborhood safety on these figures as a general ballbark. The neighborhoods with the lower crime rates are the ones I would prefer to live in.
The west end of Manchester has its problems but there are many neighborhoods that are family friendly neighborhoods. I lived in one of those neighborhood for a few years. That particular area was just south of St Anselms College to the Bedford border.
There is an area to the east of Manchester..such as off Wellington Rd that is a better area, as well as the far south side and of course the north end.
New Hampshire is a very safe state relatively speaking but we do have a few larger cities. These cities are like any other cities in the country. Crime happens. If you want to live in a city, you learn and research which neighborhoods you would want to live in. That goes for any large city no matter where in the USA you are.