Netplay

SNES

RetroArch

•  Snes9x or bsnes cores.

GGPO

Arcade emulator built around netplay.

Dolphin

Official Netplay Guide

If you don't have a very powerful CPU you may want to not even bother.

Note: The developers are working on integrating Nintendo Wi-Fi Connection to connect emulators to Nintendo's own servers.

PCSX2

Unofficial Netplay Build

Netplay has never officially been added to PCSX2, but someone tried making their own branch a few years ago, only about seven fighting games work and the project has been abandoned.

Note: You can connect to Sony's own servers though: How To Play PCSX2 Online (See Online PS2 Games for what games still have servers up)