I'm a fan we are currently in the process of AI with one of our mares and it is the ideal way to breed for us. You have more chance of it taking as the vet can track seasons and/or administer hormone injections to find when the mare ovulates. You also end up with a much broader range of studs to chose from as you don't have to travel either horse to be covered. It is also a lot safer for all involved especially the mare. The mare we are currently breeding is a sweet mare but wouldn't be best impressed with a natural covering so AI is the safest option for us :)
Its handy for barns that cannot house a inbreeding stallion. Also, its less stressful on mares who are too shy to approach a stud (I am being literal, some mares are so submissive and scared of studs - especially loud, bravado studs).
Also this helps when multiple mares cycle differently. Your vet can wait for the exact moment the cervix is loose and inseminate, so increasing odds of the made catching.