Fill the red area w/ some river rock, gravel, or fill rock. That corner has been missed so much the dirt off the pavement is just gone and useless. Gonna keep getting worse.
Do something like put down railroad ties or a field rock and mortar wall - not too tall - where the outer black line is kinda located. Then fill with decent dirt to where it is level with the bottom of that rock wall already there or the height where the inner black line is, and put in a bunch of appropriate plants/flowers/weeds.
Sacrifice a few square feet of yard space, add a nice border that defines where they definitely don't want to drive, add some color.
Oh, and tell the damn dog to move - its his fault if the drivers have to go off the road to avoid hitting him
