plug the ethernet-cable into your USB-to-Ethernet-adapter.

make sure the ethernet-cable is connected to your modem router.

open the lid on the back of the base station and unplug the power adapter-cable.

So, follow these steps to get it to work : There is no WiFi-module inside the base station.

Even when docked in the base station, it is the Switch tablet that connects to WiFi directly. If the tablet isn’t inside the base station, the Switch can’t access the wired ethernet-connection, as there is no wireless connection between the tablet and the base station. To use a wired ethernet-connection on the Nintendo Switch, you need to keep the Switch tablet inside the base station. There are 2 sides to this problem, and you might even have to take care of both :

But the USB-to-Ethernet-adapter that was working perfectly on our Wii-U isn’t even recognized by the new Nintendo Switch. I would prefer to connect our new Nintendo Switch gaming console to the Internet using a cable, as I did with our Nintendo Wii-U.

Usb 2.0 to ethernet adapter hardware