The vast majority of hostels that I've stayed at during my travels in Latin America have had broadband and WiFi. In fact, that's one of the criteria I use when I select a hostel. Having WiFi for my laptop makes all the difference in the world. Most hostels also have a computer or two for the travelers to use.
There are also many locutorios in almost every city. These are Internet Cafes and usually also have telephones. You use the computer or telephone (you can usually call anywhere in the world) and then pay at the front desk on your way out.
It just occurred to me that it might be useful to share what Internet speeds I've encountered during my trip, so I'll post the ones I remember here (not many) and I'll update this post as I think of it.
- Buenos Aires, Argentina (download) — 3Mbps
- Salta, Argentina (download) — 1Mbps
- La Paz, Bolivia — 650Kbps / 116Kbps
- Tupiza, Bolivia — horribly slow
- Lima, Peru — 450Kbps / 140Kbps
- Quito, Ecuador — 460Kbps / 122Kbps
- Managua, Nicaragua — 440Kbps / 170Kbps
- Puebla, Mexico — 1Mbps / 180Kbps
- Durango, Mexico — 786Kbps / 95Kbps
To test your Internet speed, use Speedtest.net. It's great.