Find a large log and split it, the wood inside should be much dryer than the outside.
Look under rocks or foliage for dry pieces of wood that should be better for fires.
The underside of wood (facing down) should remain fairly dry so long as the wood sits above the level of any flowing water.
You should set a shelter, tarp or other dry material to shield the fire spot, create a small lifted base from rocks or wet woods and start the fire on top of this.
Create fire as per usual.