You can get by with a 1500 sized vehicle - we haul a two horse bumper pull with a crew cab hemi Dodge. Just be careful.
For ultimate stability, you can't beat a gooseneck and a 3/4 ton though. We have a canvas top stock trailer, sometimes haul five horses, saddled up, in it, and yes, you can feel them move, but it never feels hairy... unlike the 1500 and the two horse bumper pull.
Keep in mind its not just towing capacity or braking, but the suspension. A 3/4 ton has beefier suspension so your truck doesn't get squirrley when the critters your towing shift their weight around.
"We are here to laugh at the odds and live our lives so well that death will tremble to take us."