Unless the house is desperately short of storage, I'd leave it as you found it. Put in a closet and it won't be particularly safe to access. A decorative lighted niche or two perhaps.

Display shelves

Bookshelves or wine bottle storage.

You can't mess with the remaining framing

Inserting shelving between the framing (no sheetrock, all wood)

Check Pinterest for "storage under stairs" and there is a lot of ideas.

shelves!

You don't say where the stairs are located within the house, but that's the kind of space my father would turn into a storage area... he'd use it for seasonal things or holiday decorations, so he might add shelves or not, but he'd build doors for access from the stairs (and to hide whatever was stored there).

Thanks for everyone's thoughts...

The stair are centrally located in the house. Although they are directly in front when entering through the rear(primary egress) door, they would not make suitable storage for clothing as they are not large enough, perhaps hats and gloves only.

My thoughts are to infill the sloped chimney with bricks to make it a clean vertical brick surface and to finish the framing with sheetrock or ship-lap creating a rectangular nook. I considered installing an outlet and phone jack to make it a landline phone station, cell phone charging station and possibly a central spot for the wifi.

My fear is it will just become a convenient spot for clutter.

If you do brick infill, you will never get a good match for the brick and or mortar and it will stand out as a patch job. If it were me, I would trim out the whole thing in wood so everything matched. Do a trim board/panel to match the slope of the slanting wood brace and covering the brick edge. Make it uniform in width (it will cover more brick at the top than the bottom), and put in a small, triangular wood panel at the bottom of each of the two sections. This will give you two rectangular sections to use as niches, book shelves, or whatever. It will look much more finished and natural than something that has a mismatched brick patch.

My my one year of food storage. Hiding Christmas presents. Hidden cove for valuables.

Christmas tree storage!