There's a lot of ways to do this, and I'm not sure there's a wrong way. The green beans will climb whatever you put up. You just need to make sure that once they get bigger that the bamboo stakes will be able to support the combined weight of all the bean plants.
If it were me, I would give each plant something vertical to climb. If the only support your plants have in the middle is a horizontal string it will be a little more difficult to get them to go up on their own (depending on how close together your horizontal strings are). So, I would run some vertical strings too...going from the top to the bottom.
Again, if it were me, I would probably build a lightweight wood frame and run horizontal and vertical strings across it every 5-6 inches.
That being said, people sometimes tell me that I tend to go a little "over-board" when it comes to garden utility structures, so you're bamboo stakes may fine as you described.