The Phrase Finder says that this expression refers to “A narrow-minded or unprogressive person; one who lacks initiative. I’ve more often heard it used as a put down for someone who isn’t willing to take part in some activity because they think it might be unwise. I recall circumstances where I thought the “stick in the mud” was taking the responsible approach. Regardless, it originated with earlier versions of “stick in the briers, clay, mire, etc. that date back to the mid-1500s. Being stuck in mud was referenced twice in writing in 1733.