You need to put some sort of solid cover on the deck caulking will not work as the deck boards move all the time.check out dek-master it is a vinyl covering like flooring but madefor decks . First you will have to install plywood over what you have and then the deck-master with an adesive .

It sounds like you have grading and leakage problems and hope that you receive helpful advice around sealing the deck in order to resolve the issue. I have a feeling you may always get seepage between deck and house and mold will develop in your basement, which is a worse problem. Is there a way to remove the boards, correct the drainage, and reseal the foundation? Another idea may be to add an awning that extends from the roof of the house and covers the deck, so that rain cannot filter down the sides of the house to under the deck. Good luck.

Caulking will be a waste of time. There are some good waterproof membranes out there and although they are not cheap it will keep your basement dry. Consider it a cheap way to gain useable square footage. I like fiberglass decks and there are also weldable membranes but the fiberglass is very durable and completely waterproof. There are also fluid applied membranes but I worry about them getting punctured easily. It will be critical to prep the area properly and you may also want to slope the new deck to avoid puddles and water seeping back towards the house. Spend a little extra to do it right and it will last longer and keep your basement dry so you can use it without worrying about having the contents or any work you do to the basement from being damaged which could cost you far more.