I plant runner beans in a pot and I use 4" x 2" wire fencing. I build a cage 8ft high and 6" larger in diameter than the pot. Do the same with my Tomatoes but keep the cage height to 5-6 ft.
Jim

Runner beans are not self clinging nor do they have tendrils like peas. They climb by twirling round and round a vertical string, pole or stick. They would be happy going up a mesh fence but you might need to guide them occasionally. They can get up to 12 or more feet so I hope you have a high fence, unless you don't mind a tangle at the top where they wave about and grab each other in a search for support. And don't forget they will fruit both sides of the fence, so unless you have access both sides you may miss some beans. If this happens they might stop producing. With runners it is important to keep picking for prolonged production. If you do so they can crop for three or four months.