Correcting the problem may allow for grass to grow in the affected area. Adding an extension to the base of the downspout will carry water away from the site. The use of a perforated pipe will help to diffuse the amount of water. Grading the site so that water runs away from the downspout should help as well.
In the spirit of making lemonade from lemons, the area at the base of the downspout could be turned into a small pond. This could be purely decorative or serve as a reservoir from which to water plants and the lawn. A small pump can help to retrieve the water.
If site alteration is an unrealistic solution, you may need to abandon the idea of grass. Planting a rain garden or selecting ground covers, trees and shrubs that are tolerant of excess moisture may be the best solution.