Sorry, this is in poor taste. Does this reference the fact that one with “the runs” must run to the bathroom, or is a consistency issue (i.e. “runny”). Just curious. Someone in here must know.
it could possibly be a combination of both…
Interesting question. Unfortunately I cannot answer it. There is, however, a similar Swedish dialectal word rännaskit/a with a similar meaning, and when I look the word ränna up I find that it not only is equivalent to eng. run. It can also have the meaning of flow, so my guess is that it has to do with the consistency. The lexicon I looked it up in also mentions a couple of synonyms.
Just had to be Floater to make a reply on a subjecy like this 
Considering that this state is also called “the trots” (as in trotting back and forth to the little dopers room) I would guess the origin is from that, with “runs” having the witty double-connotation.
An Irish specialist in post-gallectomy-belly-problems wrote a paper entitled (roughly) “Post Colisystectimal Diarreha - A Running Commentary”… I would love to know if it was deliberate.
The vulgar English term could simply be a translation of the more clinical, Greek-derived diarrhea, which literally means “running through”. Given the “through”, it’s more likely that this latter term refers to the feces rather than to the act of running to the bathroom.
My favorite quote ever from a textbook: “If diarrhea is the runs, dysentery is the squirts.”
Medical microbiology was a fun class.