Preventing Clogging/Flooding at bottom of exterior basement steps
Same sort of basement entry here in Seattle. Intermittent flooding and the hassle of sweeping wet leaves out of the stairwell for too many years.
  I love my current solution.
A friend drilled support-heavy screws into the concrete foundation.
We cut 3 pieces of heavy duty plywood to cover the space and fit around the bases of the railing at the top. One end of the wood rests on the screw-supports. The other end rests on the concrete on the other side of the stairwell. (3 pieces because they are heavy - so the dog would not crash through if he walked on them.)
Rain rotted my first try, so I bought sheets of that rippled Fiberglas roofing type material and cut them to size.
If you get the picture, you’ll be able to figure out how to keep water from getting into the wood through the seams in the Fiberglas.
The stairwell stays dry all winter.  I can store lawn chairs and such down there.  The cat can go there and sleep.  I can still access the stairwell if I need to get out fast by crawling out or lifting the plywood from below. If you don’t have animals, you might be able to use just the Fiberglas sheets.
PS make sure this cover slants so the rain runs off. My screw supports are just a little under the siding.