Look at where the stain is peeling away. If it's below the sink, or along the base of the cabinets where they're bumped or wiped with a damp mop, the problem is water getting under the finish and making it peel. Consider the possibility of sunlight degrading the finish if the stain is only fading and peeling away on areas that are exposed to sunlight through a window.
Scrape a little of the stain. You may find that it's not the stain itself that is peeling, but the layers of varnish over the stain. If the stain color of the wood matches the surrounding color, but you can scrape off flecks of clear or slightly yellowed finish, the stain is probably fine but you must re-varnish that part of the cabinet.
Sand the damaged area, always sanding with the grain of the wood instead of across it. Start with 120-grit sandpaper to remove the peeling finish, and finish with 180- or 220-grit sandpaper.
Match the stain -- it may be easier to take a door and a scrap piece of beech to a paint store and have an employee custom-match it for you. Most paint stores do not charge extra for custom-matching stain or paint. Follow label directions, and apply the stain to the bare, sanded wood.
Varnish the cabinets with two coats of oil-based or acrylic varnish. Use a good-quality varnish; cheap varnishes can be softer and won't protect your beech cabinets from moisture and sunlight as well.