I had a finished basement that leaked water from three different walls and by the next morning the water was gone but the damage was done. I have called experts in this field but they want so much money to come in and dig around the walls, insert a pump and then fix what they tear up. What can I do to stop this problem without spending money that I don't have?

Sell the house and move?

Seriously... its not something that's going to fix itself, and if you waste money on half fixes that don't work it will end up costing far more than it would to get it fixed right the first time because you will still end up having to dig something up and put drains and pumps to get rid of that water.

Smoothy's right. You obviously have a fairly bad water intrusion problem. Best thing to start with is making sure the gutters are open and downspout spills at least 3 or more ft. away from the house, downhill slope.