It really is that easy, Hosfac.
Pictures for you on what I did.
First drag out the foundation, and a basement beneath it.
Next, select the stair tool you want to use and position the stairs on top of the foundation, without placing them (ie. don't Left-Click yet)
Press and hold the Shift key. The staircase will disappear. Move the mouse until you get this. Then, Left-Click to place the stairs.
Now you have stairs running through a foundation into the basement.
About the missing "walls" next to the stairs beneath the foundation. This part:
To create walls around the staircase, I deleted the foundation around the staircase.
Then I covered everything with floor tile: no evidence of missing foundation.
Using the constrainfloorelevation cheat, I closed the foundation with walls.