Lawns are tolerant of foot traffic to a degree, but constant walking over the same place will eventually wear down the sod. Stairways, whether up to the house or down a hill to another part of the yard, are a funnel for foot traffic, making the point adjacent to the stairs vulnerable, even if the rest of the lawn is holding up fine. The soil becomes compacted, strangling the roots, and irrigation water rolls right off the hard surface.
If the lawn is just starting to wear thin around the stairs, avoiding walking over the area for a few weeks can allow it recuperate. Make sure there is nothing else causing the problem, such as a clogged or misdirected sprinkler head. If the problem area is at the top of a stairwell by a retaining wall, the area may need extra water to support lush green growth. These areas dry out quickly because the drainage system behind the wall wicks away the water, a problem compounded on south-facing slopes by the heat of the sun against the wall.
If the lawn is already worn through to bare earth, buy a piece or two of sod and replace it. Most garden centers sell rectangular pieces of sod in spring and fall. Using the blade of a flat shovel, cut out the worn sod in the shape of the new pieces, loosen the soil to a depth of 6 inches and mix in a 2-inch layer of compost before pressing the new sod firmly into place. If needed, remove a bit of soil to make the new sod even with the level of the surrounding lawn.
Installing stepping-stones where the grass meets the stairs is a simple way to solve the problem. This preserves most of the greenery and provides a surface on which to walk. The area at the bottom of the stairs could be replaced with a landing, a common approach when designing new landscapes. Choose from concrete, brick pavers or flagstone to match the existing aesthetic motif in the yard. An area of pebbles or decorative gravel framed by a wood border makes an equally effective transition from lawn to stairs.