In process of making a cherry frame and panel blanket chest, 40" long x
19" high x 18" deep (approx.) Corner legs (stiles) are 1-3/4" square
(finished dimension); question is, for this length, how narrow can I
make the top and bottom rails? I plan on using 3/4" finished dimension
stock for the rails, and there would be a single panel, probably solid
wood, grain running horizontally, of approx. 38" x 12" or so; it would
sit in a 1/4" or 3/8" groove in rails/stiles. And, stiles would be
mortised to fit a 3/4" long tenon in rails. Suggestions appreciated.

I would say this is a totally astehtic question. One the panel is boxed
by the frame it is a pretty strong structure, even with a floating
panel. I would make them either exactly the same as the legs/stiles or
some derivitave adding or subtracting 1/3rd's. (ie 2/3rd's wide or
1/3rd less) This is sorta close to the golden triangle value and a
quick way to generate some pleasing ratios. Just sketch it out and see
what looks best to your eye.

One other possibility is to do a wider rail at the bottom. This is
especially appropriate if there is no base to add some visual weight to
the bottom of the piece. So top rail exactly the same as the stiles and
bottom rial 1 2/3rds' larger.