Local resident and born and raised Barcelonian here.
Have your kids grown up in an urban environment or a rural one? Have you given them a certain degree of freedom in the past? Would you say they’re more naive or street-smart?
While it’s true that well-travelled kids tend to be more street-smart, I’ve lived in the US as a teenager, and I know well that US parents are generally more overprotective than Europeans. This often results in more naive adolescents, which is why I ask.
Barcelona is "statistically" a safe city, but it's still a city—and a multicultural one, with people from many different countries. So, you get many different ways of behaving and understanding the world.
Moreover, it's one of the hottest tourist destinations in Europe. Tourists come and go, staying only for a few days, so both local and foreign criminals target these tourist spots, knowing that even if they’re caught, the tourists probably won’t return for the trial. Without their presence, it becomes harder to convict them for minor crimes. Beyond local criminals, we even have, like Paris or Rome, transient ones who travel on cheap flights to other cities in Europe to commit crimes and then return to their home countries afterwards.
This mix of factors means that, while the city is fairly safe, there’s still a certain level of crime—something inevitable when there are large concentrations of people in one place.
As Carlos mentioned earlier, just on the other side of the Gothic Quarter is the Raval, which, although not inherently unsafe, has a less 'healthy' vibe for a teenager, with some drug dealing, street prostitution in certain streets, etc. So, it might be better if they go with someone else. That's my take anyway.