Having water runoff that erodes yard.

Can you rebuild the ditch and then build a retaining wall on the side of the ditch?

What a gravel or stones?

How about a rock garden/berm? It may force the water to at least slow down. there are rain gardens that may hold water for a while

Install a drain system and plant groundcover that will grow in shade. Check local building codes to make sure you do not drain on to your neighbors property.

You have to slow down the flow, an unpaved single channel will not do that. You can pave the ditch to keep the flow open, but that will not stop the erosion. Sight unseen, it sounds like your driveway is uphill of your back yard and that the rain is running off the paved surface into the grass. Where the driveway meets the yard you could make a dry bed of stones/gravel. On the downhill side of the stones you could use pavers or blocks to retain the stones. That would interupt the flow across the yard. You could design it so the rainwater runoff was diverted into a paved channel or pipe.

I have the same problem(s)

Here is what I am doing:

Home Depot: Pavers $1.00 each.

I purchased 400.

I stacked them 2 or 3 high, starting at area(s) direction where the water is running from.

Stacked them in the direction away from your drive way.

After you have double checked, this blocks the water flow from your drive way....add flowers that comes back each year & multiples, small scrubs flowering plants etc, this will enhance the paver wall & give you crub appeal.

Below is a picture of what I started. I hope this picture will help to understand what I was talking about.

You have to do a little bit at a time & watch the flow direction of water. Lot of time I worked on it while it was raining to get it right..

Good luck, on you project. It is hard work. The pavers are heavy & about 2 or 3" thick.

A French drain should solve your problem. https://www.thespruce.com/how-to-install-french-drains-for-yard-drainage-2131133

got a picture? sound to me you could use a french drain at the end of the driveway right before the backyard to direct the water down the driveway to the street storm drain. https://www.youtube.com/watch?v=wMbO5RIji_Y

Where is all the water coming from on your drive that isn't draining into the street storm drain? If you gutters are dumping extra water consider installing extenders and drains to move it down the yard toward the street storm sewers.

Look into the possibility of a rain garden. They're easy to create, and will soak up much of the water that comes off your driveway.