Is it possible to post a photo of it? I need to picture the slope before I can offer a good answer.

agreed, photo will enable to give a better answer

It depends on what you want to do. If the slope is too steep to mow, then planting either Junipers, periwinkle, Ivy (if you have a lot of shade), or any other ground cover plant. If you want grass, then you will need a straw mat to hold the slope then seed the bank. If you had pictures, it would help.

Even if the slope is moderate...sod could be installed. Properly directed gutters will reduce runoff under the eaves.

Dig a trench at the bottom of the slope and direct the water flow away from the level yard area - probably out to the curb.