Thicker grass would be a help. Try a top-quality organic fertilizer along with water in a program designed to grow thicker grass. Don't be afraid to experiment a little. A taller height of cut might work better. Or perhaps a short cut--followed by letting it get tall.
A denser type of grass might suit the situation better--like one of the denser sports-type bluegrasses: Sydsport, Blue Velvet or Langara.
If all else fails--new sod might be the quickest.