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What is the water source? It looks like it is coming from a drain pipe behind a shrub? I would recommend relocating that to dump to a different location. No matter what type of edging you install on the bed with the mulch, a massive amount of water from that down pipe will cause some overflow and mess. Maybe you can incorporate a stone trail where the water flows, if you can't relocate the down pipe. Ideally, I'd want to see the drain pipe go under the side walk, and out by the grass, or if its by a garage, down the driveway.
You can install a flexible downspout extension on your downspout that is next to your front step and run it behind your bush out past your mulch to keep the water from washing your mulch away. You can buy the flexible downspout extensions at Lowes or Home Depot.
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