Water finds it way to the lowest point because of gravity. So if we dig a mine or something same, then how we take ground water in account, so that it doesn't fill in.
We pump it out. Open and closed pit mines usually have pumping installations to get the water out.
Look up Mine dewatering on Wikipedia.
There have also been numerous accidents in the past where mines were flooded (YouTube: Turkey mining accident: 18 workers trapped after underground water floods mine), and when left alone many abandoned mines will fill up with (ground) water.
Sometimes these are nice diving objects.
Or (as Trond Hansen commented) we try to prevent it from coming in in the first place, by using cement, grout or bulkheads to 'plug the holes'.
If you are really interested:
Mine Water: Hydrology, Pollution, Remediation by P.L. Younger, S.A. Banwart, R.S. Hedin.
Design of Water Holding Bulkheads for CoalMines by V.S. Mutton and A.M. Rememnikov
Your assumption about water that reach the lower point in mines is correct, in fact in the older mines that develop let's say in a mountain they dug a lower level tunnel that collect all the water of the mine, this tunnel can run several miles like one in Sardinia (Italy) that reach the sea at 14 Km.
This method have the obvious limitation that you cannot go under this bottom level, so you reach a point when you have to pump water out to your daring tunnel.
Mine shaft also have a deeper bottom to collect water, in older mines with a relative low water income they run a "bucket" in the bottom of the shaft to take water as in a normal well.
In the modern times you have always the need to collect water in a point where you have installed your pump, and you do this digging some ditch along the tunnel for example.
Water can be very dangerous because digging tunnels you can encounter large mass of water, this happen for example during the dig of many tunnels that go trough Alps.