Did your new verizon Wi-fi router have enabled "WPA2-PSK (TKIP)" wireless security types parameter?
P.S. Wii console dont have "WPA2-PSK (TKIP)" wireless security types.
3DS console have "WPA2-PSK (TKIP)" wireless security types.
Look below, ...
Also, ...
Error Code 51030
http://www.nintendo.com/consumer/wfc/en_na/ds/results.jsp?error_code=51030&system=Wii&locale=en_US
Error Code 52130
http://www.nintendo.com/consumer/wfc/en_na/ds/results.jsp?error_code=52130&system=Wii&locale=en_US
Error Code 51330
http://www.nintendo.com/consumer/wfc/en_na/ds/results.jsp?error_code=51330&system=Wii&locale=en_US