I'm going to assume that when you use the term "best friend" you mean someone they very often spend time with and someone they constantly confide in first or mostly. In which case, my answer is Yes, it would bother me. I do believe that in a serious relationship, your primary friend and confident of the opposite sex needs to be your SO and that's all there is to it. Everyone else needs to take a step back.
That being said, I could never expect anyone else to "change" just for me. If something about someone's SO doesn't fit in with how they believe a relationship should work, then that person needs to make the choice to stay and deal with it, or to leave. You can't expect the other party to change just because you believe things should be a certain way.
However: The term "best friend" is thrown around way too lightly I find... if you're just talking about a group of friends that maybe occasionally he hangs out with and talks to a bit here and there, then No, that really wouldn't bug me all too much as long as an open invitation was offered for me to be a part of the group.
If it's going to be some sort of "secret club and you can't be involved with my friends" sort of deal, then again Yes I would have a problem.
DH knows that I don't really think highly of having close friends of the opposite sex in a committed relationship... however he doesn't share this view, and I respect and accept that. He does however respect my view, and he has already told me on many occasions that if the situation should come about, he'd gladly offer to take me with him if he was going to go off with a female friend, and I told him that any of those friends are of course welcome in our home.