Brussels is the "de facto" capital of Europe and whether people accept it or not, it does house the main European establishments, which makes it a city at risk of terrorist attacks. Though this does not dwell on regular residents' minds, it also does not change the reality. Certainly the security measures you encounter when visiting some buildings will remind you of this fact.
There are sometimes strikes and demonstrations held in the city protesting European or Belgian law enactments/proposals. Though this represents no danger to citizens, it substantiates that Belgium a very significant city politically. Behind the calm, happy-go-lucky nature of the city, there are real issues being dealt with that affect the lives of not just Belgians, but Europeans and often the world as well.
Specific threats in the city in everyday life, in particular for women and young children, will be riding late night trains/subways. Reports of real problems are few and far between, although there is sometimes an intimidating atmosphere which is best avoided. Like any reasonably sized city, there are areas that you need to be more cautious in than others.
Generally speaking, Belgium is safe and once you become oriented, you will almost certainly feel very comfortable in most parts of the city.