I'm not sure from your post how much water you are really getting. You said pours down the stairs. That could be a lot of water.
If you do stop the water on the outside (steel door and threshold) you may have problems with water buildup.
One of my kids was living in this dump of an apartment complex (surprisingly a national company) and it had open stairs to the basements of the buildings and during a really bad winter spring I saw these stairwells almost completely full of water. When the water receded the doors had been completely destroyed. Bent in at the bottom and twisted.
I gather this was from the pressure of having all the water against it.