First of all, it's really too bad that the former owners cut down trees and shrubs before you moved in. What were they thinking?
Now to stabilize the area, we recommend grasses for controlling erosion because of their extensive fibrous root systems that serve to hold the soil in place. You can probably buy plugs of the grasses below, but the cheapest thing to do is to throw out some seeds. However, just throwing grass seeds over the side of your bank may not work very well. The seeds need moisture to germinate. If the moisture comes in the form of rain, it is likely to wash the seeds down the bank into the drainage ditch before they have a chance to germinate and take root. You might consider an erosion control blanket. The erosion-control fabric works by slowing the runoff water and allowing sediments to fall out rather than be washed away. Seeds are sown under the erosion-control material and grow up through the matting when they germinate. You can also insert plants into the soil by cutting through the matting. The roots of the plants that are growing through the erosion-control material anchor the soil to stop the erosion. If you use erosion-control blankets made of biodegrable material, they will eventually disappear leaving the plants to control the problem. Many nurseries carry this erosion control fabric. You could also include low growing shrubs and other perennials in the mix with the grasses.
Here are some grasses and other plants that should work on your bank in shade (less than 2 hours sun per day) and part shade (2 to 6 hours sun per day):GRASSES: All the grasses listed here are bunch grasses and grow to around 3 feet.
SEDGES: Sedges are grass-like and evergreen and generally grow less than 2 feet tall.
SHRUBS: These shrubs grow to 4 feet or less.
Now about the dogs—my experience with dogs is that they like to dig in soil that has been freshly worked; therefore, I would recommend that you temporarily fence the area until the plants are well-established. Otherwise, you are likely to lose a goodly portion of your plants. Hopefully, after the plants are well-established, the dogs will be less eager to dig in them.
Here are photos of some of the plants above from our Image Gallery: