My turn to weigh in ;D:
Let's get one thing clear first: I think the Hall of Fame should be based almost completely on on-field performance. I think the Hall of Fame selections are there to honor the greatest players in the game (managers, umpires, pioneer/legendary executives, too), and the museum in Cooperstown is for fans and students of the game, not the whimsical guardianship of Major League Baseball or, heaven forbid, the BBWAA.
IMO, the "integrity, sportsmanship, character" clause in the HOF voting is, at best, irrelevant and, at worst, hypocritical. There are too many persons voted into the Hall for whom this clause is completely overlooked, for popularity reasons or otherwise. And frankly, it doesn't matter anyway. As fans, we overlook all kinds of questionable characters as long as they help our teams win -- hell, just look at Pete Rose, the player. The teams overlook this stuff, too. Look at a situation like Brett Myers -- accused of beating his wife, yet allowed to start the very next day. Yes, off-the-field character matters much to MLB ... NOT. So it shouldn't matter to the Hall of Fame (which, fyi, is a separate entity from MLB -- MLB does not run the Hall or make its rules. The Hall could reverse its 1991 rule not allowing "ineligibles" to be elected and MLB couldn't do anything about it.)
You need more examples that "character counts"?
Ty Cobb (also an avowed racist, who may or may not have killed a man in self-defense/cold blood) and Tris Speaker almost surely fixed a late-season game in 1919 between their two teams (along with Joe Wood), yet it was hushed up by Ban Johnson when it came out years later, then absolved by Judge Landis, and the two (still playing at the time) left the game quietly, but without punishment. They're both in the Hall ... and rightfully so. (Johnson and Landis are in the Hall, as well.)
Leo Durocher (not exactly standing at the door of integrity and character, either) hung out with gamblers all his life, was suspended for the entire 1947 season because of that, MLB likely had evidence that he was betting on many sports (baseball among them? who knows?), and he's in the Hall ... also rightfully so.
There is much more evidence than not that points to Jackson (who was actually a fairly decent man, personally) playing his absolute best in the 1919 World Series, despite accepting $5,000 for the "big fix." There is no debate about that: Jackson did take the money. Did he earn it? That's part of the debate, but not the important part.
(I'm not going to get into the details of the "Eight Men Out" story here, but the crux of my argument would be this: Judge Landis' punishments were the same, even though the players' crimes were not. Landis, btw, had a reputation for getting his outlandish decisions overturned when he was a judge in federal court. But he was free to do whatever he wanted in MLB, so, hey, if he wants to ban just eight of 'em for life and overlook other players who had done the same (fixing was fairly prevalent in the game then, but it was also an accepted part of the game), he could. And if he wanted to overlook the owners' complicity in trying to pull a Richard Nixon and go out of their way to cover up the big 1919 Series fix for a year until Cicotte finally confessed anyway, then he could do that, too.)
So my opinion is: You can't keep the best players in the game out of the Hall of Fame. That does a disservice to the game and insults the public's intelligence who realize that a Hall of Fame without the all-time hits leader and the third-highest batter (and we'll see what happens with McGwire when he becomes eligible in 2007) is not really a legitimate Hall of Fame at all.
If MLB wants to keep Pete Rose out of the league completely, banning him from any coaching or managing or even instructional or promotional job in the majors, they've got every right to. Hell, they've got every reason to. Ban him from the game, that's fine. I have no problem with that. He broke the rules and he should be punished. Punished for life, even. Don't let him earn a living off baseball, as they govern it. That's perfectly acceptable.
But the Hall of Fame is not about MLB. It's about the greatest players in the history of the game. And Pete Rose and Joe Jackson are among the greatest players in the history of the game -- and really, there's no argument against either of those claims. They should be in the Hall.
If you don't want to have anything about them in the museum itself (also separate from the actual Hall of Fame), as more punishment, fine. If you want to add an asterisk at the bottom of their Hall of Fame plaque with "banned for life" or "indicted for steroid use" or some such designation, go right ahead. But you can't deny their place among the game's greats. And you can't deny them a place in the Hall of Fame. That's insulting to any baseball fan who watched them play. They were Hall of Famers -- put them in the Hall.