Mark:
I played (although I haven't in a year) for years. The best places to go play basketball are usually at clubs--it's hard to find pick up games. I had been going to club Pedro Echague in Flores. They have an open gym night on Wednesdays a year ago at around 9:30 for an hour or two usually. It's mostly older guys (25 and up, lots of guys older than 30 and some in their forties) so don't expect a super high level. Pedro Echague had two TWO wood courts, unheard of in Capital. It's a small club in Flores, and I think they charge 25 pesos to go for those two hours. Once you go there, you can probably get invited to outdoor tournaments. I got involved in one in Bajo Flores right outside the villa there. It was pretty interesting.
Also, I did play some pick up when I lived in Belgrano in a huge Polideportivo in Colegiales about 10 blocks or so from the train station in Colegiales. I think you can find the polideportivo on google maps. During the summer, people start showing up at around 7 and they will play 3 on 3 or 4 on 4 depending on who is there. It's pretty random as to which days people show up.
You can try a club, but most of the time those are just practices which get guys ready for the tournaments. You can usually practice, but that sort of sucks because there are drills and all that (would just rather play myself). You can't play in the club tournaments because you are a foreigner, unless you have citizenship. They did this because some guys that played in college would come live here for 3 months, help some tiny club get from the B league to the A league, and screw up the club system, then leave (and the club promptly goes back down to B.
Most clubs don't have open gym nights for basketball players, but if they do, ask for when the "veteranos" go and play (the people not on the club team). If you don't say veteranos, the secretary you talk to when you call will have no idea what you are talking about and will tell you to go to a practice, and then you will find yourself sitting in the bleachers watching a practice. Veteranos is the key word here.
Pedro Echague is one of the most recognized clubs in the city as far as basketball goes. It's a tiny little club de barrio, but they have an first division, second, and third division team, the newest court in the capital, and lots of socios. People there are very friendly. I would start there if I were you. Feel free to send me a message if you want more info.