They are sagging because the weight of the shutter has crushed the soft white pine tenons over the last 113 years. There is no perfect way to fix this. A cabinetmaker of the old school could add new slips of wood to the tenons to firm the joints up.
A quickie repair would be to epoxy the tenons and mortises together while clamping it square, but this is a last-gasp irreversible remedy.
You could try screen door turnbuckle braces on the unseen side of the shutters, as it will hold for a time and does no real permanent harm.
Casey

The thing with the turnbuckles is that you will have to adjust them over time as they loosen--I've had to do this on the one on my rear screen door. Also, if you decide to operate the shutters, the turnbuckles will show when they are closed.

It may be a chore, but if they are already separating at the joints, I'd go ahead and add small shims to the tenons as suggested above.

For the second option mentioned, rather than epoxy, what about a wood filler instead?

Thanks for the replies. The joints are not separated enough to allow room to access the tenons.

Is the idea of using flat L brackets on the back to support the wood considered just too amateurish? Someone previously attached this board to the back of one of them, and then screwed a few of the louvers onto it.

I repaired some absolutely hideously massive shutters (closed they cover an 8x10 doorway), and added heavy L and T brackets. It did work, but only because I mortised them into the wood and the shutters are under a porch and get no wet weather. I was also able to use a lot longer screws because these particular shutters had frames 1 3/4" thick.
Casey

It is hard to imagine that no one makes solid brass flat L brackets. The best I can find is galvanized steel.

Brainerd Hardware is the company I knew of that made all sorts of solid brass hardware.

Thanks for the tip, millworkman. Brainerd is now a subsidiary of Liberty Hardware, and my search of their site is coming up empty. Sigh.

Look for 'campaign hardware' and you should find what you are looking for.

That was helpful. Brickeyee's search suggestion led me to Horton Brasses, which has the flat L brackets in solid brass, used as truck brackets. They are 1/16 inch thick. I need to contact them and see what they say.