The left slot on the adapter (while engraved "Wii U" facing up) is the first port, and so on, so don't mistake it for being the other way around!
#Before Dolphin 4.0-4599
The GameCube controller adapter is not supported in versions older than 4.0-4599. Updating to a newer version is necessary.
#Dolphin 4.0-4599 to Dolphin 4.0-8593
With any necessary installation complete, run Dolphin, and check Direct Connect under GameCube Adapter in the Controller Settings, then restart Dolphin. Set GameCube ports to Standard Controller corresponding to each port on the adapter that is plugged in with the actual GameCube controller. The setup for the adapter is now completed. To test, simply start a game that uses GameCube controllers.
Dolphin supports hot-plug, so the pads can be unplugged and plugged into other ports. If a pad is unplugged from the adapter, Dolphin will use what is configured (PC keyboard/mouse, PC gamepad etc).
With any necessary installation complete, run Dolphin and click the GameCube controller dropdown. Select GameCube Adapter for Wii U on any slot that you wish to use the adapter for. Hitting Configure when this is selected allows you to turn rumble off/on per controller, as well as a toggle for using the DK Bongos.
Unlike 4.0-8593 and before, you will have to choose between GameCube Adapter for Wii U or Standard Controller for a real GameCube controller or the device emulating a GameCube controller. The hot-plug properties are also removed.
It's worth noting that only the first instance of Dolphin Emulator will detect the adapter. Another instance will detect the second one if inserted, and so on. If the Dolphin Emulator couldn't detect the adapter, it's possible that you already have another one running.
Dolphin might not show the correct status of the adapter if Direct Connect is unchecked. Check that then restart Dolphin.
If the adapter is not detected at all, then it may only be a matter of changing USB ports and restarting Dolphin, or its driver/setup might have recently been changed/overridden. You should follow the instruction again from one of the sections for your system.