Call a structural engineer to find out what the structural issue might be. You just described a foundation problem that needs to be addressed.

And check all of the usual suspects for moisture penetration such as the ground being graded away from the home, gutters not clogged and directed away from the home, doors and windows properly flashed and caulked, etc.

Second to what hollysprings advised.

Possible explanation would be there is an excess water source along the foundation. The basement entrance wall is the weak point. To successfully solve the problem will require finding and correcting the excess water source, then repairing the gap. That will mean exposing the outside(excavating the dirt) and reattaching the wall or filling the gap. The gap cannot be successfully repaired from the stair side.

Thanks for your quick responses! I have been looking for an excess water source, but so far no luck. We have had a lot of rain lately and the leaves are starting to fall so I'm sure that's not helping, and although the eaves are not empty, they aren't full either.
I'm not sure that I was completely clear. There is no water that penetrates the inside. The wall of the stairwell ends at the wall of the foundation, like a T intersection. The water is finding its way through the gap where the stairwell wall meets the foundation wall and is pooling at the bottom of the stairs, on the outside of the basement. Does that make sense? Does that make a difference to your response?

Thanks again for your help.

You seem to be missing the point.
Water cannot be "held back" or it will find a way to create even more damage. The description of the stairwell abutting wall space opening up wider at the bottom than at the top is also a possible concern. You likely need to extend your eaves water runoff further away from the foundation and also determine if the "water rush" is the result of your weeping tile (if equipped with same) being blocked from draining normally elsewhere as it should. At best, you need to confirm your weeping tile is draining correctly or can be easily fixed to work right before dealing with your wall problem.

Thanks laat. I got the point that water cannot be held back. Hollysprings and handymac suggested a structural engineer was required. I wanted to know if that was still their recommendation or if there is some other professional that I should call or avenue that I should pursue? The eaves will overflow if congested with leaves, but water does not drain anywhere near the foundation, so that is not the issue. I'm just trying to understand my options.
Thanks for your input!

Water will take the path of least resistance so it doesn't matter how far away the water source is located. If the soil was loose when your home was built the water may have formed a path back then. Over time it could have eaten the dirt that is under the concrete at the bottom of the steps and things are now starting to tip into a hole. Once water that is running towards your basement wall comes in contact with it, again, it will take the path of least resistance going downward by going between the dirt and the wall.
Plan on having to seal the basement outside wall, replace the drain tile that should be near the bottom of the wall, and maybe even install a new staircase.
A shop vac attached to 10 or 15 feet of 2" pvc pipe that has a couple 90 degree units attached at one end should help with keeping the gutters clear. Just hook it over the gutter, and vacuum away the leaves and water.

An experienced landscaper might have the knowledge to abate the water, but a soil expert/architect will have the knowledge to fix the rest of the problem.

As has been said, thee should be a drain outside of the door at the landing of the stairwell to get rid of any water that makes it to the stairwell. You can patch the gap with hydraulic cement (it expands when it cures) but this won't make it watertight. In fact you should leave ~1-2 in. of the gap at the bottom so that water behind the wall cna drain into the landing area fo the stairwell and the it drains however the drain works.

We have the same issue whereby the stairwell (added later) has settled and pulled away. I graded the soil above the wall. The water that still comes through the crack goes to a floor drain, so I don't worry about it too much. I haven't bothered to patch it, since any further settlement would just reopen it, thought aesthetically I suppose I should.

Perhaps injecting with urethane would solve it, as they do with some foundation crack repairs. Urethane allows flexibility for any further joint movement (up to 50% of the thickness of the crack).