Reply by MattD

I think CutNRun is right. 1/8” hidden or maybe even through dados if you want to show the joinery.
Another configuration you might not have considered would be to use a single vertical divider. One advantage to this is that you could perhaps put in a door in place of the 2 bottom left drawers. I studied a lot of post and rail pieces like yours years ago and 3/4” is quite standard for horizontal dividers. However, what are you going to do for the drawers fronts? flush, overlap? I suggest you think about how you want the space between drawers to look before you decide on how thick the dividers should be. Just a suggestion. Looking great so far!

-- Matt - Syracuse, NY