The Sinister Six has rarely existed as an entity for any real length of time. Generally, they have been pulled together to achieve a short-term goal (usually, but not always, involving taking down Spider-Man). Generally, one of the reasons why they don't continue to work together long-term would be that they are often captured while trying to accomplish said goal.
In-universe, I believe the longest-lived version would be the group assembled by Doctor Octopus as a part of the plot that culminated in the "Ends of the Earth" story (AMAZING SPIDER-MAN 682-687). That team consisted of Doctor Octopus, Electro, Mysterio, Sandman, Rhino, and Chameleon.
In our time, they seem to have been working together about 19 months (from AMAZING SPIDER-MAN 648 through 687 - the title was published twice a month during this period). While they were not seen in every issue, they appear regularly (and even showed up in other titles, like AVENGERS ACADEMY), always furthering Doc Ock's scheme.
- Following the "continuity implant" that had Chameleon and Kraven the Hunter having a long history well pre-dating their initial meetings with Spider-Man, Chameleon's presence in the Six can be seen as a direct replacement for Kraven, perhaps explaining (out-of-universe, at least) why Doc Ock has included him in the team in the most recent incarnations he's assembled (even when a resurrected Kraven has been available).
- For about 15 years (from around 1984 to around 1999), Sandman's participation in the group was generally against his will - he was at least going straight, and frequently an active hero during that time frame.
- That said, it took over 25 years for the team to get together again after AMAZING SPIDER-MAN ANNUAL 1 (summer of 1964) - the "Return of the Sinister Six" storyline in AMAZING SPIDER-MAN 334-339 was published bi-weekly in the summer of 1990.
Technically, the grouping seen in SUPERIOR FOES OF SPIDER-MAN was around at least that long. This version (with Boomerang, Shocker, Overdrive, Speed Demon, and the latest (female) Beetle) only had six members in their first outing (with the Living Brain, in SUPERIOR SPIDER-MAN #1), but from their until the end of their own series would be about 22 months. That said, none of them are likely to be who anyone would think of when calling out Sinister Six members.
However, the longest lived variation on the theme is almost certainly the Sinister Syndicate. This was a group of villains organized by the original Beetle, and they were together for about 5 years (from AMAZING SPIDER-MAN 280 (September 1986) through DEADLY FOES OF SPIDER-MAN mini-series (ending August 1991). With team members Beetle (original), Speed Demon, Rhino, Boomerang, Hydro-Man, and (later) the Shocker, this was basically the direct ancestor of the "Superior Foes" version of the Sinister Six (in fact, Shocker recommended the name, but the others felt the Syndicate had been losers).
Overall, looking at the memberships of the various versions of the Sinister Six (and, again, discounting the "Superior Foes" version and the Sinister Syndicate), Electro was probably the most consistent member of the team. Interestingly enough, he's also the one who created his own version of the team (the Emissaries of Evil) to fight Daredevil in DAREDEVIL ANNUAL #1 (from the 1960s).
From there, the "Ends of the Earth" team (in-universe) are the ones who've served under the name longest, with Mysterio and Sandman having participated not only in Doctor Octopus' version, but several of the other versions of the team. I would personally be inclined to replace the Chameleon in that roster with the Vulture. In spite of my note above, Chameleon is not someone who's been associated with the group historically, and the Vulture was also a part of most prior versions of the group, with or without Otto.
Note: if you want to take the Syndicate into account, then Rhino is probably the one character who's served the longest, being both part of the Syndicate, and the "Ends of the Earth"-era Sinister Six (and in the newest version, seen in the Miles Morales title, SPIDER-MAN 234).
So, I'd say that:
- Doctor Octopus
would probably the 6 characters with the longest tenures as members of the Sinister Six. That's basically the original roster, with Rhino replacing Kraven.