Be very careful adjusting pots on arcade monitors.
1) They may be old and brittle, one wrong turn and game over.
2) They're not really designed for continuous adjustment, they're really designed for initial adjustment and then to be left alone. Set and forget.
I need to warn you as I don't want to be responsible for any damage to your monitor. Sometimes things are best left alone, if it's working don't try to fix it philosophy.
That said, if it appears fine with a signal I wouldn't touch a thing. If you are seeing these lines when a signal is present, then yes, turn down the brightness pot. This should be all that's necessary, if not then you will need to locate the 'screen' pot on the chassis and slowly and gradually turn it down until the lines disappear.
The screen pot is like the overall adjustment range. It dictates the range of the brightness pot. Think of it like having a sound pot on an arcade board (the screen pot on this analogy) and the volume control of your amplifier (brightness pot). If the sound pot is turned down fairly low, the amplifier volume control will have a limited response. Conversely, if the sound pot is dialled in to the maximum setting, the amplifier volume control won't have much range, it will begin very loud at the lowest point and get extremely loud with every little turn upwards. So the two pots need to be adjusted accordingly for an acceptable range of adjustment.
Hope that makes sense.