How do I stop water from wicking into the wall?
To answer both questions will require to have the concrete step removed entirely. 
The water is somehow migrating up to the floor through capillary action I suppose.

To check it and fix it is still the same remedy. The step needs to be removed, downspout, as well as the siding and the corner boards. There are joints in the corner boards that will make it a bit easier to do so. No need to remove the complete corner board. Remove all the trim and siding below the door sill and in the corner that the concrete covered. 

Since I have been in WA state, I have seen, typically how siding is installed under doors and steps, and this appears to be no different. In MD, it was never a good idea to place concrete directly against the wood siding, or any siding for that matter. This invited rot and insects due to harbored moisture by the concrete. I don't understand why it is done this way a lot over here, but it seems the norm. Perhaps it the condition of the trade everywhere, but i only seen it here.

Back at the matter. Under the siding there should some continuous barrier, galvanized metal is used a lot out this way, with a lot of nails holding it in place. which is the problem, holes in the face of the metal. This is what will need to be replaced. Even if it is not galvanized metal, this is the part that is failing, exacerbated by the steps.

The replacement should be a "peel and stick", self sealing membrane that covers the whole run of siding and trim that the stairs covered. It needs to go higher than the floor level on the inside, at least by a few inches, and around the corner to meet the siding under where the downspout goes back on and seal the end with caulk to the siding and place the corner board while the caulk is still wet, to bond it all together. The trick will be the door sill, sealing that. Typically the membrane is in place before the door is set and the bottom is sealed with caulk as the door is set in place. The steps can be re-poured, but they really should be kept away from the siding to keep the trapped water problem from reoccurring.