You have to be careful when calling basement contractors because they'll all want to sell you their solution. I would call someone who can repair the damage to the walls (a mason) and ask how they would solve the problem - or at least what's causing it.
Meanwhile, take a look at you downspouts - are they draining away from the house? Do they drain into a pipe at the foundation? Are you sure it's not clogged?
Also take a look at the way the earth slopes away from the house in that area - make sure it actually does slope down and away from the house so that water is not coming toward your house (like from the neighbor's downspouts.)