Basic/Expert/Etc D&D: There is the Known World but to me Karameikos is the main part of that worth trying out. The GAZ1 supplement is pretty good if you really want to go with the Grand Duchy as a home nation, but the rest of the world never did a ton for me. I saw it as an example more than a real world. If I wasn't doing a homebrew for my new Basic campaign I would go with my 1st edition choice which is...
1st Ed AD&D: Greyhawk, hands down. None of the others match up so well with it IMO. The history, the locations, the names - there's no need to retcon anything to fit in strange new races or classes. You don't need to adapt anything.
2nd ed AD&D: Forgotten Realms - this is what I played in for most of 2nd edition and a lot of the specialized kits and specialty priests really end up fitting the Realms well.
3rd ed D&D: Scarred Lands (from Sword & Sorcery) I would guess that Eberron is a good fit here too, but I've never played it, run it, or read the books, so I'm leaving it out. I have run a nearly 2 year Scarred Lands campaign though, and it feels nicely different, but still D&D. Intersting monsters can be taken from the Creature Catalogs, much flavor can be found in the region books like Mithril or the Blood Sea, and the Player's Guides have a ton of interesting background and prestige classes. It's clearly built for 3rd edition rules and it shows in every aspect of the material.
4th ed D&D: Not sure - I haven't looked at a lot of the campaign material that's out there and my plan is to use a prehistoric version of Greyhawk when I do run it (as in "before history" not "lots of dinosaurs running around"). I know they mixed up the realms material quite a bit, and they haven't published anything on Greyhawk yet, and no "new" world has come out either. It may be Eberron by default here, being the newest of the campaign worlds.