New York City
Hello - I will be staying in Buenos Aires and Santiago, each city for 2 nights only.
In Buenos Aires you could stay in Recoleta, as mark said, or in Palermo. If you are near the D subway or avenue Santa Fe it's really easy to move around by bus and taxi also. And these area beautiful neighborhoods too.
Mentioned in this answer:
I think that the best place to stay in Buenos Aires is Palermo. It is pretty close to everything and is a really good district for nightlife and restaurants. There are plenty of things to do locally and there should be easy access to the metro.
I never had an issue in cabs (single male) however you may need to be prepared to pay a bit of a tourist tax as they will generally take you the long way around. If you don't mind walking there are lots of things walking distance to Palermo and it is safe enough to walk around during the day.
Hi! Porteña here! First of all, I personally wouldn't go on cabs alone in Buenos Aires... you never know, and the city isn't exactly renowned for its safety. If you're going to use public transport (which is incredibly cheaper) you're gonna need a SUBE card. You can see where to buy it and recharge it here: https://www.sube.gob.ar/MapasSUBE.aspx?id=1, and I advise getting the Moovit app.
I think this http://www.turismo.buenosaires.gob.ar/en/article/barrios exposes the most can't-miss places/neighborhoods in the city.
As the others said, Recoleta is probably your best choice.
For BsAs I recommend Recoleta if you plan to see sites. Stay near the cemetery and there are restaurants in the area. Easy enough walk downtown.
The other great area is Palermo but that is more for dinner / drinking. Can always take the subway over quickly if you get tempted, although typically the subway closes down before you are done being out. I have never had bad luck there with a taxi, but have heard they can be shady.