You need to divert water flow away from driveway, or install permanent raised edging along the sides of the driveway. Obviously, this cannot preclude airborne dust and dirt from settling on the drive, so if your environment has these problems, daily sweeping or washing of the drive will be necessary.
Hi Mike, hope this video helps
How is it in the concrete driveway? is it from flooding somewhere in your yard pushing dirt onto driveway? Need to dig trenches along sides of driveway to direct water flow where you want it to run. Is it coming across bottom of driveway along street ? Need to create trench along street that enters spillway into drain under drive. Really could use a picture I have way too many questions is it happening often
Install a border on either side so dirt from flower beds does not wash onto it. For a cohesive look, add the same border to the flower beds.