Hi Alex. If you have a small PCB veroboard already made up then changing to opto isolators wouldn't be the right thing to do at this stage. I see you can fit this veroboard inside your hand controller.
From what I can see of the handset tracks. It looks as if the white or black wire is ground...but this needs to be measured with a multimeter. Any cheap jaycar meter on DC volts will work.
However...you may not need to connect the ground..from what I can see.
ST4 is "earth switching". It provides an "earth" command (or switched earth....aka a ground pulse) from the guide camera for the relay coils pin 3, 4, 5, 6.
This energises the appropriate relay for what ever the pulse duration is.
So wire it up without the ground and see if it is driving. If you are concerned with not connecting the ground then measure the wires to find ground on the handset and wire the pin 2 ground to this same ground.
Clear as mud?
edit:
and to correct the above image I posted for ST4 earth switching...this would work the other drawing would not because ST4 is earth switching not Vcc switching...sigh.."oh Magoo you've done it again!"