We have been in Barrio Norte, near Santa Fe and Coronel Diaz, for over 12 years now, and it is safe, nice, and friendly. I know many of my neighbors, I know for many years my local vendors, shoe repair people, fruit stand, restaurants.
A very good mix of ages, and incomes, mostly apartment buildings, which weeds out the Mercedes SUV owners and the really rich, who can often be pretty crappy neighbors.
Other neighborhoods I would consider would be Belgrano R, Collegiales, Villa Crespo, and Caballito.
I have (argentine) friends who live in all of them, have been to many people's house in those areas, and found similar small scale, friendly neighborhoods. I would avoid Palermo Soho, which is mostly bars and boutiques these days. Palermo Hollywood, all the way over to Dorego, is seeing massive development, with two story houses being torn down and 8 story apartment buildings being built, its moving very fast. The area near the mercado pulgas, which used to be very quaint and quiet, is all yuppies and artisan bread and coffee bars now.
Across Cordoba, into Villa Crespo and Chacarita, is the up and coming hip neighborhood, where its still a mix of old funky with new artist run hip spaces, but still reasonably priced- we end up going there a lot to eat, to see music, to see galleries- its another place I would look.