Try to use less stain or finish when going near your edges, and if some drips over the edge immediately wipe it off.

Each board has a show face and a not-as-important face, like the underside of a table or inside of a drawer. Apply stain or finish to this face first, then flip it over and work on the show face.

If you can flip it over, place it on something with a small point, like these painter's pyramids. If it does leave a small dimple it's really hard to see. If you can't flip it over easily, let the first side dry before staining/finishing the other face.