This answer comes directly from the Nintendo website:
Error code: 52131
The Wii cannot connect to the Internet
Possible Cause: The wireless router is not connected properly or the Wii console will need a DNS manually entered.
How to fix: Check how your router is connected; manually assign a DNS to the Wii.
Check connections to your wireless router:
Even if you can get an Internet connection to your computer through the wireless router, if it is not properly connected to the modem, the Wii and other devices may not be able to use it.
Most routers have five (5) ports on the back. There is usually a set of four, with a single port set back from the rest or separated by a different colored border or line. This single port is usually labeled “Internet” or “INT”. Make sure the cable coming out of your modem is plugged directly into this “Internet” port, not one of the other four that are grouped together.
Manually change your Wii console’s DNS settings. To do this, follow these steps:
Go to Wii system settings (how to), double right arrow and select “Internet”.
Select “Connection Settings” and choose the connection file (1, 2 or 3) you are currently using.
Select “Change Settings” and click the right arrow 3 times to access “Automatic DNS Obtain”.
Select “No” and choose “Advanced Settings”.
Click on the “Primary DNS” field and type: 8.8.8.8
Click in the “Secondary DNS” field and enter: 8.8.4.4
Note: These are public DNS provided by Google and are subject to the Google Terms of Service. If you prefer not to use this service, searching the internet for “public DNS” will give you several other options you can use.
Select “Confirm” and then “Save”.
Select “OK” to start the connection test. When testing is complete, try downloading the update again.
If you still receive this error code after trying the steps above, please call 1-800-255-3700 for assistance.