There are a few things you can do to reduce the numbers.
1. Shoot the little creatures when they present themselves. 17 HMR or 22 Mag fit the bill. They will not spook the horses as bad as a hot centerfire.
2. Burrows away from barns and buildings can be smoked. Little Giant Destroyer or similar "Smoke Bombs" work well. I tape 2 to a dowel about 18" long, light them, wait for smoke and stick them in the burrow. Cover with dirt to keep the smoke in the hole. I do this in the late evening to make sure they are in the burrow. This is best done in the spring thru fall. Wintertime is not as effective since the 'hogs place several earthen plugs in their burrow to protect themselves against fox, coyotes, etc from digging them out during the winter.
3. Trapping. I use Safeguard cage traps, 11"x12"x30" or 11"x12"x36", are the 2 sizes I use. Place the traps near the entrances to the burrows. If you can "funnel" them into the traps, even better. Use boards, cement blocks, 1/2"x1/2" welded wire (hardware cloth) to create a fence to force the 'hog into the trap. Make sure everything is secured and tight. Giving them no option other than entering the trap will guarantee increased success. Other good sets include setting up on well used trails. I have caught 'hogs, with no bait, by placing 2 traps end to end on a trail so that no matter which way they approached they would be greeted by an open door.
220 body grip traps also work. Set them over the entrance to get the 'hog as they come or go. To make them 'hogs only, make a 18"x18"x18" box out of 1/2"x1/2" hardware cloth. Leave one end open. Set the 220 and place the box over the burrow and the trap. Secure the box to the ground. Now any 'hog inside will get caught, but non-target creatures outside are less likely to get caught because of the wire box screen.
Cage trap bait. I use apples, pears, melon rinds, strawberries, broccolli, zuchini for eye appeal. Strawberry puree lightly trailed into the trap and place behind the trip pan works great. Do not put the puree in front of the pan, since the 'hogs will dig. Daily's mix in the bottle works great. There are also commercially made baits out there for groundhogs, and when used properly they all will catch 'hogs.
Feel free to contact me if you hae any questions.