Choose groundcover plants that are native to your area to keep from planting plants that will become invasive.
Lay landscaping mesh over the problematic area. Overlap the seams of mesh by 2 to 3 inches. (Landscaping mesh will allow groundcovers to spread while biodegrading over the course of a couple years.)
Read your particular plant's spacing requirements on the nursery tag. Cut small X shapes into the mesh to mark the locations of the plants, using the spacing recommendation.
Dig holes into the soil under the X shapes, making the holes as deep as the root systems of your plants. Dig the holes with a garden trowel.
Unpot the plants by gently pressing on the pots until the plants release. Set each plant's root system into one of the dug holes. Cover the roots over with soil. Pat the soil around the base of the plants to secure them in the ground.
Water each plant with a light mist until the roots are saturated. This will take a few minutes but will help keep water from eroding the area further. Water the plants this way as often as recommended on the plants' nursery tag.