The picture of the wet basement is the result from the water softener continually backflushing, thus super saturating the ground. This caused seepage through the wall. The water softener has been stopped since and what you see here is the result, a dry floor, even when it rains!
Gary hasn't been able yet to identify the softener's make so he can't test it to see what's wrong. Hopefully, he won't get rust stains on his clothes in the meantime!