Do desalination plants affect the ground water table?
With desalination plants the import thing to consider is from where such plants are taking water and where they are disposing of their waste.

The waste from desalination plants is either salt of very salty water. If the plant you are referring to is just dumping its waste on the ground nearby or into a nearby river or water channel it is possible that the salt in the discharge water from the plant will enter the ground water, making it salty.

Generally this should not happen and in most situation it does not. With desalination plants near the seashore, the waste water is discharged back into the sea, but far from the intake for the plant.