Go to Almagro!!!
Look, im a young woman living alone, and im easily scared/very wary of unsafeness, and i live in Almagro in where i easily get back home walking at 4 am, i take buses at every hour, the atmosphere is amazing, never had any problem whatsoever, and its the safest place i ve ever lived in.
Palermo is nice, Recoleta is nice, but this is where pickpocketers go to find people with money. Alamgro is RIGHT next to this two neighbourhoods, but it haves a lower profile, so there are no pickpoters or bands of young guys looking to steal something....nothing! The part between Corrientes and Cordoba is the best one, a lot of movement of young people and families at night,people walking their dogs at midnight, etc....it is THE BEST neighbourhood in Buenos Aires in relation to safety, and this is told by someone that lived in other neighbourhoods before (San Telmo, Palermo), and someone who is a woman too.
Plus, it is cheaper than Palermo, San Telmo, Recoleta or any touristic area. It is closer to everything (Almagro is the more convenient neighbourhood in terms of location, all the lines of buses, 2 lines of subway, etc), it doesnt flood,it haves great atmosphere! Im shocked not everyone is screaming "Almagro" in this thread.
Dont go to Belgrano: it floods.
Go to Almagro, BELIEVE ME, moving here is the best decision i ve ever made.