I think Northam should resign unless he can show he had no connection to the blackface/KKK yearbook photo (which seems improbable). However, I do wonder about the scope general principle at work here. It may not matter for Northam's case. But it has obvious implications going forward. This is not the first controversy of this kind, and it almost certainly won't be the last. Here are some issues worth considering:
Perhaps anyone with a demonstrable history of racism should be drummed out of polite society permanently. The obvious problem with this is it destroys incentives for penance and self-improvement. Historically, every successful struggle against bigotry (and other evils) has been won in significant part by converting people who previously supported the evil in question. Even temporarily stigmatising all such people may be problematic, if only because there are so many of them.
Maybe the principle should be limited to barring such people from offices that wield great power (such as being governor of a state!). But they can pursue a wide range of other careers and should generally be accepted in every part of society, so long as they have sincerely forsworn their past behaviour. If so, there's a question of where we draw the line. But I nonetheless have a lot of sympathy with this position.
Are the principles discussed in 1 and 2 limited to histories of racism? How about other prejudices, such as homophobia or sexism? What about people who were apologists for brutal but (at least in theory) non-racist tyrannies, like communism? People who publicly defended the likes of Hugo Chavez and Maduro (some still do!)? Depending on context, these other forms of moral obtuseness can cause as much harm and/or offence as expressing racist sentiments can.
Maybe the answer to 3 is we should only stigmatise people if they express awful views that are widely condemned in society at the time, across the mainstream political spectrum. Thus, being a communist sympathiser should be stigmatised today, but not, say, in 1940. If Northam had done what he apparently did in 1950 (when blackface was far more socially acceptable) rather than in 1984, maybe he should be excused, as well.
Another possible distinction is that stigma should be limited only to those who knowingly promote evil. If they did so out of well-intentioned ignorance, that qualifies as an excuse. Thus, wearing Che Guevara clothing is OK if you don't know what a brutal mass murderer he was, but not OK if you do. Notice that, by this standard, Northam could be excused if he was not familiar with the history of the KKK back in 1984 and/or the racist implications of blackface. If you think such historical ignorance is implausible, consider all the people who still think that, e.g., the Civil War was not about slavery, and that the Confederacy was not established for the purpose of protecting the "Peculiar Institution."
Whatever you think should be stigmatised, can there be any way for the people in question to "rehabilitate" themselves? Can they even do so to the point where it is acceptable for them to be in a position of great power? If so, what do they have to do? Is it enough to admit the error of his or her ways and promise not to repeat it? If not, what more is required?
Some will look at these difficulties and conclude we should never use social stigma to punish people who express awful views. On this view, the relevant distinctions are too hard to draw, and in any event the way to deal with such people is by persuasion, not ostracism. That is not my view. I think stigma can play a useful role, and at the very least it makes sense to exclude adherents of some types of awful views from positions of great power. But the issues listed above are, I think, far from trivial.