What is normally done to bridge skirting boards around an opening with different floor levels on each side?
Dimensions on your photo would help make a better model, but here is what I might do in this situation.
The easy way
End the upper trim with a bevel (30-45°) at the edge of the stair, then continue in the lower room as normal.

Transition
You may want to use a transition piece to pull the pieces together.

You could even cut an angle into the top of the transition, for a different look.

You may also want to bevel the upper trim into the transition piece.

The complex, lots of angle calculation, pain in the neck way
Cut a small angled transition piece from the upper trim, down to the lower trim.

In this case, you may still want to bevel the upper trim piece.


There's not really a right way to do something like this. It's more about your preference, and the amount of work you're willing to do.