I have found that starting a fire in the damp in not too hard, but the fire needs to be tended to otherwise it will go out and you will need to start it up again. When it is damp, the bigger the dry tinder bundle and the more kindling, the better. Then split and feather up tons of wood before you start the tinder going. I would even go as far as saying to have your wood pile set up to dry next to the fire before lighting up to.

Even if it is not raining, raise the wood pile off the ground (on a couple of pieces of wood will do) and cover the top of the wood pile to keep condensation off. I raised the wood off the ground, but I didn't cover it last time because it was nice and clear and sunny out, but because the area was sooooo green, the dry wood turned to soggy sticks even in the afternoon sun.