We have two kids (8 and 13) and live in Belgrano. We send them together across the street to the supermarket to buy stuff. Of course, I sit on my balcony with my sniper rifle overlooking the street. NO NO NO - Just kidding! I would not have a problem putting my family in any of the neighborhoods here except near a vila (Retiro, etc) or in high tourist area (Recoleta, Plaza San Martín, etc). I think if you stay away from those two areas you are pretty safe.
Getting your purse, wallet, backpack, etc stolen is likely to happen here at some point. It's a reminder to be more vigilant and tune your radar to your environment.