Board games with games of chance? Never. Board games like DnD? Maybe. However, it would have to be streamable. I think the entertainment value isn't there because it isn't associated with many button presses. I can see Monopoly streams being big. Sometimes chess, but in the realm of eSports, they seem off limits by discrete technicality.
I could be mistaken, but I do believe a lot of popular board games do have competitive leagues. I think they are more geared toward the tournament flow than leagues.
Although I doubt esports would see them as they are not an esport. Perhaps one day there will be well-made video game versions of popular board games and they will take off, but until Hasbro and the like stop using the video games in a mobile style, half-finished, microtransaction machine, we will never know.