It is very probable that you heard some scary stories of stolen backpacks and wallets before you even came to Barcelona. And after arriving you have probably heard some more. Even if you are from here or have lived in the city for many years, you always know you should watch out for some classic problems.
The problem Barcelona is the most famous for is pickpocketing. You should keep in mind though that this is not just aphenomenon happening in the center. Looking after your purse a little bit more than at home is therefore not enough. If you want to make sure you leave with the same bag you came with, you should not leave your belongings alone in any public place, and this includes most shops and other safe-seeming spaces. Avoiding thefts may seem self-evident but the level of pickpocketing in the city is so high that avoiding it often means a substantial change in habits for those not used to it.
The second problem some tourists face is cheating regarding taxi fees. Make sure you do not pay any supplements, unless you are sure it is needed due to for example luggage or point of pickup. In general, the taxis of Barcelona are reliable, though, so this is not a problem you need to stress about nearly as much as pickpocketing.