The more I look into this idiom, the stranger it sounds to my ear. I think it’s because of the use of to come, rather than to become; I can’t think of many other phrases where one just comes something.

Anyway, its origins are interesting. Meaning ‘to fall over or fail at some venture’, come a cropper is first recorded in the mid-nineteenth century. It is a figurative elaboration of the eighteenth-century phrase neck and crop, which referred to falling off a horse.

A horse’s rear end was – it maybe even still is – called the croup, from the Old Norse krupp for a protuberance (giving us at least one sense of the word crop). There is also a crupper, the strap on a horse’s harness that passes back from the saddle under the tail. A small whip used in horse riding is called a crop, but the origins of this don’t seem to be related.

Essentially, if you went neck and crop, you were off the whole horse, as in…

The startish beast took fright, and flop
The mad-brain’d rider tumbled, neck and crop!

So if you come a cropper, you’ve fallen off your metaphorical horse and you’d better get straight back on and either shut your stable door before it bolts or lead it to water and try to make it drink.