Take care of the water issue if the wall must be repaired due to water problems. Often water leaks aren't caused by damaged pipes, but are simply due to loose fittings. When you find the source of the leak in the pipes, unscrew the connector from the pipe with a wrench, dry the pipe with a towel and wrap six to eight layers of pipe tape around the threads on the end of the pipe. Once the pipe tape is in place, screw the connector back in place tightly with the wrench.
Cover the sink and the countertop with plastic, and tape the plastic down around the edges. This Old House recommends painter's tape for securing plastic.
Press the drywall back against the wall, and nail it in place to the joist if the wall does not have cracks or holes caused by water. Apply primer to the area where the wall was wet and the water stains remain. Paint the damaged wall with a paint that matches the wall's color to blend it in with the rest of the wall.
Fill small holes or cracks that are smaller than an inch with drywall compound until the compound is even with the wall surface. Once the compound dries, use sandpaper to make it perfectly even with the wall, and paint the area to match the wall.
Place mesh tape over any holes in the wall behind a kitchen sink that extend beyond 1 inch. Center the tape over the hole and press it in place. Apply drywall compound over the tape, allow it to dry, sand the compound to roughen it and apply a second coat and then a third coat. Feather the coats so they blend into the wall. Paint the area to match the wall once the compound is completely dry.