Watch out for pickpockets on the major shopping lanes. Especially in spring and summer there are quite a lot of pickpockets around. Then stay away from the part of the city where the Russians live (Falconplein and neighbourhood). Some cafés and bars are rather 'strange' and 'dark', if you know what I mean...
Like all over the world in big cities, beware of pickpockets, especially in the shopping area's around 'De Keyserlei' and 'Meir' and the historical old city. There're a lot of beggars (geapsy women with children) don't give them anything. They're not so poor as they look!
Beware of street children (mostly of Romanian origin) whose 'loving' parents send them off to pick pockets of careless tourists. But generally Antwerp is a safe city. Just use common sense and that's all.
Beggars can be very annoying. I don't have a problem with the ones that just sit there, but some gypsy women and children can get very pushy. You might feel sorry for the children, but I still wouldn't give them money, as you'll only encourage these people to abuse their children that way.