I've had no luck tracing an origin for the use of "skinner," but a couple of observations on the comments above are in order.

A large percentage of the comments relate to use of the term "mule skinner" in the western states. It should be noted the the question came from a song about the Erie Canal - and from a "back east" location and tradition having nothing to do with most of the West.

Several dictionaries I have consulted uniformly define a "skinner" as one who "operates a team of draft animals." The term appears to have been in established use well before the period(s) cited in the western US, although I have yet to confirm this.

The term "skinner" was used in relation to most kinds of draft animals, although by the early 1900s, calling someone a "hoss-skinner" was, apparently, considered something of an insult - at least by breeders of draft horses like my grandfather! "Ox-skinner" was commonly used in my area (apparently with no negative connotations) as late as 1948, although the use of ox teams was rather rare by then.

I will note that my grandfather, who was a die-hard draft horse breeder and user, had great respect for mules and those who used them. While he was quite outspoken about people who were "too dumb to deserve a good horse," he also had some respectful words for those who chose to take on the "extra effort" that is required to get the best out of a team of mules.

quoting grandpa:
"Any d... f..l can drive a horse. Too many of them do."
"A good mule can outwork any horse, but you've got to learn to talk mule to get them to work."
"Nobody can drive a team of chickens."

I think that last one applies to herding cats.

My grandpa's main objections to breeding mules were:
You have to keep a donkey around.
There were too few "customers" smart enough to use one.

Incidentally, our nearby Amish-Menonite community at Yoder, KS stil has their annual mule races and rodeo. Quite a show.

John