Seed will just remain dormant until the conditions are right.
I would address the errosion concern by installing silt fences, hay bales and spreading straw. You really can't put too much down, but if you don't put enough out, you'll end up with some nasty gullies.
You can buy rolls of burlap with seed in it that is staked to the ground. It's pricey, but very effective. You anchor it down with metal clips so it holds the soil in place during a heavy rain. Then when conditions are right, the seed will sprout. In time, the burlap will rot away and you are left with growing grass. I was told that all the highway projects are required to use it here in Texas. I don't know that for a fact, but I have seen it used quite a bit. I looked into it to protect my dame after my pond was done, but went with just broadcasting seed and then filling in the ruts from the rains until the grass filled in.
Remember, you have to slow the water down, and spread it out. As a general rule of thumb, it always takes twice as many barriers then you think to get the job done. If going with a barrier, like a silt fence or hay bales, run it across the top, then again half way down. Dependingon the size of the slope, you might need another line or two.
Silt fences are pretty cheap at the box stores. Just be sure to get the material all the way down and into the ground. If the water flows under it, then it's not doing anything. I try to get down as far as possible and then add a little dirt to the fence to help create a dam. In the low areas, I try to put something on it to keep it down. Bricks or chunks of concrete. I've even used logs.
Eddie