Home Inspector Missed Something....?

If the area wasnt wet during the inspection he probably could not tell. It is worth calling. Take photos of the mess, and the areas leaking and keep them as evidence. Take photos of the area outside close up and pulled away to show the conditions outside. Photos are always good to have for proof. Continue doenting the entire winter to show mold growth, ruined carpets etc as time goes on. Your best bet is to go to Home Depot after you call the builder and if he will not come and fix anything, then get prof advice and seal up the best you can from the inside to avoid further damage. Photo your work too. Have supplies ready for any sunny days that you can attempt to alter the area outside too. I honestly have rarely been in a basement they are not common in my area. But my thoughts would be, pull back any carpet and padding and get a fan running to help dry the area out. Buy those crystals they sell that you place ina bucket to absorb humidity. See if you can find a sort of spray on styrofoam to seal up any areas you know are leaking. But Home Depot really has alot of trained people to help with home repairs or any other local hardware store. Keep all your reciepts as you may wind up taking the builder to small claims court. So give the builder the first option, you might luck out and have an honest and honorable person to work with.