We have an electronic scoreboard for our air hockey table that doesn't work any more. We tested the AC/DC power supply and found that it was putting out almost 12V, while it is listed for 6v 400mAmps..

Can this have burned out our electronic scoreboard?

What can I look for on the circuit board to see what can have burned out for failed?

 

Unregulated wall warts often have no load voltage that is much higher than they are marked, but 12 V from a 6 V supply seems exceptionally high. If the wall wart is the original that came with the table, try measuring the voltage output with it plugged in the table and under load. It will probably show close to 6 V. If so, it's not the cause of your scoreboard failure.

If you can get to the control PCB for the scoreboard, post some pictures and maybe someone can spot a problem, or at least suggest how to check it.