It seems to me that what you're asking isn't just specific to Firefly. It strikes me that you're essentially asking how to run a campaign based around one or more crews of spaceships. Generalising even further, you could essentially say that the question is how to structure a campaign based around one or more self-contained groups of PCs, with the intended focus being on the varying adventures of said group(s) of PCs. I haven't seen Firefly, so it's possible that I'm not aware of some aspect that makes it unique and unable to fit with the usual organisational styles, but so far as I can tell it is do-able.
I don't see that there's anything wrong with going with a small bunch of PCs and following them around, complemented by a large crew. Moonbright takes this approach right now, and it seems to be working pretty well. Even expanded out to full weekends, I don't see that this is a bad approach, unless you're specifically wanting to have a high number of players. Dreams in the Witch House did it successfully (albeit with slightly higher player numbers at first). The one major downside to this is that I can't really see any particularly good way of determining who should get to be the players if there is over-subscription.
The second approach you suggest, contriving reasons for multiple crews to meet, doesn't seem all that far-fetched to me. If the crews all have a mercantile focus, then it is highly plausible they will all head to the same locations where the chance for profiteering is good. Since those locations are also likely to be war/disaster zones, crews with a humanitarian focus would also fit. A mixture of the two would provide for some immediate conflict. It would still be entirely possible to have recurring NPCs, in the form of other crews, members of the distant authority who are resentful at having to be the token response to these incidents, antagonists who develop an enmity specifically towards the PCs (perhaps due to events at the first game) and so follow them around, etc. In every game most NPCs are shallow one-offs anyway, and in my experience some of the recurring NPCs come from someone's backstory, which will usually give the GMs a ready made reason for them to show up in-game. Alternatively, some particularly influential NPC might send out some sort of missive to crews, directing them to meet at differing locations at appointed times. Perhaps the reasons behind this are a major element of an ongoing plot, with the motives and reasons of that NPC becoming clearer as the game goes on.
The last possibility you mention, described as Babylon-5-with-cowboy-hats, whereby you have various groups and they repeatedly come together at a specific location for whatever reason(s), is also a valid option. It is exactly what Crucible is doing at the moment, and to a certain degree (though it wasn't quite the same) is what Teonn did. It perhaps doesn't give the specific feel that you are seeking in a Firefly game, but there's no reason it can't be done for any given sci-fi game.
Well, this is what it all looks like to me, as someone who has never tried to GM a campaign (or indeed even written a one-off). There probably is a way to get what you want that I just can't see, but the three options outlined above all can definitely work. If you're wanting specifically the idea of the game being the adventures of a crew, then I think barring discovery of another superior method, you would need to use that first option of the few PCs with large crew.