I'm no GT expert... but I'll take a stab.
Make sure you know how to adjust the volume in the menu. Roll the trackball left to reduce the volume setting, and press the start button to save the setting.
There should also be volume up/down buttons inside the coin door. Try them as well.
If those don't work, I'd recommend repairing the PCB. I don't know of schematics for any of the umpteen GT games are available, nor if this is a common problem with a common fix.
If you decide to wire in a volume control between the PCB and the speakers, DON'T use just any little pot. As this is the amplified audio output, you'll need something that can handle several Watts. The volume controls on early 80s Atari games were wired like this. The 3W 25ohm rheostats that Radio Shack sells (271-265) might do well. I think the old Atari ones were 50ohm.
Connect the two wires from the game to the outside two lugs of the rheostat; connect the speaker wires to the wiper lug and either of the others.