Ideas which do NOT change the campaign style:
Write up a half-page recap of the story so far. Hand out copies at the beginning of every game. Read it aloud. Only write down the stuff important to this adventure, the way that tv serials explain the vitals in the show's first 30 seconds. The hand-out can also list, in bullet-point fashion, norms of play and other important information. You can email this to players before the game, but no one will read it.
Recap only when it comes up. Alternatively, instead of recapping at the beginning of play, you can just play as if everyone is in the loop, and then have brief expository asides when a player runs into something that she should know, but doesn't.
Have a pre-game warm-up. Invite the new players over a bit early. You can catch them up without wasting the time of the people who are clued-in. Make sure they have characters, know the back-story, understand the play style, etc. A brand new player could attend a warm-up session that's totally outside the normal game structure--a one-shot to get to know them, test them out for compatibility, get them used to your funky house rules, and so on.
Because I hate to lose these wonderful ideas... Ideas that entail changing the style, which C. Ross has said he doesn't want in his edited question:
Run a Western Marches-style game. Let the players who show up determine what the night's entertainment is ("we should go explore the Barrows of Delight!").
Run delves. Like the popular D&D Encounters phenomenon, your games can be nothing more than some barely-connected combat encounters. I suspect you want the story to play a more important role, though.
Make your games more episodic in style. Make each night of play a self-contained little adventure that doesn't count on too much back-story. Sure, the back-story is there but not knowing it won't break the adventure.
Use group-owned, pre-generated characters. If the characters belong to the group, you can assign them at the beginning of play. This will help with story continuity. Bob can't make it, but the warlord is still there.
Run one-shots. If you're willing to give up epic play, run single-night games. They can all be set in your campaign world, all hinge around the same plot or themes, or they can be totally disconnected.
Run one-shots of different games. Really, there are a lot of great games out there designed around getting a full-on gaming experience out of four hours of play. A lot of them are weird little indie games, but they're hella fun. If your player group is churning, and it feels like slot after slot of convention play, why not treat it as such? Break out In a Wicked Age one night and carry: a game about war the next night.