Pretty negligible, if I understand your question. My Avalon [batch 2, black case] barely effects my network's access to the Internet. Also, the reverse seems true as well. I am able to stream Netflix, browse the web, FTP large files, without effecting the Avalon's performance from what I can tell. I'd bet any DSL-equivalent Internet access speeds are sufficient. I have cable access at about 6 Mbps up and down, but would trust a 1 Mbps up/down link to be more than sufficient for mining.
_theJestre
-just to add to what Jestre said, as far as I can understand it, the latency of the connection will have a much greater effect on your performance than your bandwidth. Bandwidth requirements are pretty much negligible, whereas poor latency could cause you to submit more stales... I think...