Durham is not Singapore and crime is generally something to be more aware about here. That said, there is no need to be overly intimidated and general precautions should keep you in good stead. As always, it is better to be safe and sorry, and until you have a good enough understanding of the local areas, I would recommend keeping in mind the following points:

• On-campus is generally very safe. However, as in Singapore, you should be wary of crimes of opportunity by keeping your rooms locked when you are out and not leave your valuables unattended. There have been cases of unattended laptops getting stolen in libraries, or in the middle of the night in unlocked rooms.
• Off-campus (Central and the Gardens included), I would recommend not walking around alone after dark. If you need to go somewhere, go in a group or get someone to drive you.