“Open” may be the redundant modifier, since it seems rare to find a “sleigh” that isn’t already open (including Santa’s, of course). But “one-horse” is a necessary modifier, because there are many different multi-horse sleigh arrangements, including timber, lumber and other heavy cargo sleighs, as well as the one-horse variety (optional two-horse varieties as well) for light personnel carriage.
In that case I suppose that “one-horse open sleigh” was used as much for the meter and rhythm as it was to allow people who had zero experience with sleighs of any kind (such as most of us) to form a mental image picture of what it might look like – and how it might feel.