Pictures of the area may help.

Our wooden upper deck stair footers are inserted into our concrete patio

Yes, we need pictures. Are you saying that wooden footers are anchored in cement and that the top of the cement is below ground level?

Thats not going to be the issue with water getting in your basement and pouring something in there like hydraulic cement will not help. Do you have gutters and leaders? Show a pic with the entire elevation as you have something else going on somewhere.

We only see water when it rains, and we have 6” gutters, but we had horizontal rain and now I’m seeing moisture. The third photo is directly above where The water is coming in.

The main hole I’m concerned about it one directly into the foundation to the right of the stair photo, next to the white stucco. I can take a wider shot when I’m back home.

Patching the hole if that is truly the point of entry will be a minor fix if it works at all. The real issue is the foundation is allowing water to pass thru it. How old is the house? You may need to have some selective excavation/demolition done and attempt at waterproofing the foundation.

Yeah, definitely not interested in excavation, as it would destroy the entire patio, and cost a fortune. It’s a 1960 house and we have done a ton of waterproofing/new gutters/grading, etc, so this is literally the only place we are getting water and it’s very minimal. I’m sure the stairs aren’t up to code, but we will be rebuilding them at some point, I would actually prefer a temporary fix for the moment.

Squirt in some expanding foam caulking.