I lived in Caballito for years and I have two small children. Caballito has two large parks, Parque Rivadavia, and Parque Centenario. Parque Centenario is if I am not mistaken the largest in the city.
This year I moved out of Caballito. I couldn't take it any more. The traffic, pollution, and noise was absolutely stifling. Most buildings in Caballito are apartments, so forget having a back yard or much green there. Housing can also be expensive in Caballito compared to other parts of the city, although I did own an apartment that was too small for a family of 4 (60 sq mts.). I live in a suburban area now and I couldn't be happier. Yes, Caballito is a nice neighborhood. A good friend of mine lives there in a large apartment with an adolescent and pre-adolescent. They seem to like it, but they have been there forever. It is a nice neighborhood with so much to do and see. But it is a very stressful neighborhood with a lot of movement. I personally do not think it is good for a family with small children and that you should consider a neighborhood with more houses that is quieter. It also depends on what part of Caballito you will live in. Do not live in Primera Junta area, as it is dangerous, dirty, and the noise is unbelievable. Parque Rivadavia is a bit better, but lots of noise, too. The Pedro Goyena area is really pretty and also much quieter.
Overall, Caballito is a safe part of the city, depending where you are at. I walked all through the neighborhood and never witnessed anything shady going on, except as you go down Jose Maria Moreno and get down towards Parque Chacabuco. Caballito is not next or even close to any villa, and there are not many neighborhoods in Capital left that have that distinction.
On the plus side you have lots of restaurants, things to do and see. But you will have lots of people even for simple everyday tasks, like purchasing goods at the store. During weekends, the parks are virtually unusable because of the quantity of people that frequent them. During the day, they are nice and not so full (except for Parque Rivadavia, which is always full of so many people). There are also some "hidden" plazas that are much less crowded and are really basically like parks, like Plaza Giordano. These have less people since they do not come from the surrounding neighborhoods of Almagro and Palermo (who have less green). You do have some big clubs as well like Ferro and el Club Italiano where you can take your kids to activities such as swimming, etc.
But do not expect anything to be easy. It's like pulling teeth. I just couldn't take it anymore.