My 1830s bank barn was used for a dairy business and therefore housed cows on the concrete floor. I was considering clearing out the concrete and digging down approximately 1 foot in order to convert it to a stable and run in area for horses. Will that affect the integrity of the barn's foundation? Thank you for any suggestions.

I didn't consider that. Is that easier than taking a jackhammer to the concrete?

It's very likely your barn has some sort of "rubble" foundation. Is it possible to leave a piece of the slab in place, around the perimeter, to reinforce the top of the footing? You'd likely need a concrete sawing company to saw around the perimeter, lots of jackhammering in any case. An opinion from a local experienced builder could be very valuable. Like Derek says, you need to weigh your options, costs & benefits to choose your best option.
Good luck
Scott

How about taking some of the floor out above?