You might try filling most of the gap with backer rod--it comes in various sizes, then use some concrete patch to fill in the surface.

Thanks Columbusguy- we'll look into that!

This is the type of sealer that I see that our local Target store uses for expansion joints in the sidewalk and walls.

Here is a link that might be useful: Sealant

Thanks! Adding it to the list of "things to buy" :)

I really appreciate the help.

This is a very common issue that arises when concrete stoops are built separately from the house, typically with no compaction of the underlying soil. For decades now, new home builders have avoided the problem by just delivering the home with a couple of pre-cast steps that many homeowners replace with more elaborate landscaped steps.

In short: there is no permanent cure. You can remove them altogether, then either: excavate to undisturbed soil below the frostline and lay a proper footing and foundation and poured steps, tied into the existing foundation; or prepare a compacted base and have one-piece pre-cast porch or multi-piece pre-cast steps installed, as in the pic below.

Everything else is a cosmetic stopgap.

If replacing the existing threshold is on your list, you might want to see if an additional one could be extended over the intersection between the sinking stoop and the house.