OP - If you're not watching the dog closely enough to keep it from falling down the stairs, I'd suggest you investing in a crate, or leashing the dog to you. A dog that small shouldnt even be playing on the stairs.
The awning would not work in my case. As my stairs do not have several posts, like the one pictured. There's only two posts which are attached to two different steps. Not to mention a couple of my dogs would chew the tie-straps off.
The gate works well for me. It also keeps new fosters on the tile downstairs, while we are potty training. My upstairs is carpeted.