Hi Vagabondita and welcome.
For one, I find BA safe enough compared to U.S. cities. With a young child, we are not out late at night much. Nighttime may be less safe, I don't have any experience.
I think BA is a great city for kids! Daycare schools (called Kindergardens here) cost 20% of what they cost in the U.S. When choosing where to live, I suggest looking at schools and parks (play grounds). It's very nice to be able to walk your child to school and to the playground.
Belgrano is nice and a lot of people choose to live there. That said, I did not find the playgrounds to be good in Belgrano. While there is good housing, newer buildings, a lot of it is more residential with less to walk to. We live in Recoleta and just make the long trip to China town for shopping (which we plan to do today in fact!).
The area around the park "Plaza Armenia" is nice (a bit expensive) has a great playground and more English-speaking expats than anywhere else. Also look around these parks; Plaza Teniente General Emilio Mitre, Vicente López y Planes Square, Las Heras Park. All these have great playgrounds and family centered life.
Send me a private message when you arrive. My wife's friend has a 20% discount at the best grocery store in China town (Asia Oriental). You could go along and stock up. Make sure to get WhatsApp on your phone - that's what people use here. You can put me in touch with your husband and I could help him along with other things that can be difficult to navigate there.