Are you asking about leasing out a horse you OWN or leasing someone else's horse?
I have done both, with absolute success, and am totally comfortable doing so again, under the right conditions.
I fully vet anyone wanting to lease a horse I own, and I expect them to fully vet me before allowing me to lease their horse. I check vet, farrier, dentist, trainer, neighbor, etc. references. I drop in in-announced for surprise visits. I want to see the "real" picture, not one where they have had time to "tidy up" everything.
90% of the horses at the barn I manage are privately owned and free leased by the riding program that is run at the barn. Some of the owners are absentee, and others are present multiple times a week. Our lease program is very different from others, as we pay 100% of all feed, farrier, chiropractic, dentistry and vet. We do have a "cap" to emergency vet fees based on each horse's value in the program. The owners only pay if an emergency vet fee exceeds that cap, otherwise, they have no financial responsibility to the horse.