I am not the best with wireless and wired network management connections.
However I will give you my best go.
Open your Network Manager and look for the option to set the Wired connection as the default. (the first choice)
That way your Linux Mint will try your wired connection first and than it will choose WiFi (I think) if wired fails.
You might even be able to edit your /etc/network/interfaces file and add an argument to that file so that wired will be permanetly set to go to
A more experienced member with the knowledge of the /etc/network/ interfaces file would be the one to get advice on for this. Sorry I don't know more.
This is if you want to set a static ip address:
If that doesn't work:
Try a Google search "How to set wired connection to default linux mint"
Hope that helps.