Part of the problem is the increase in the number of users. And there's not much you can do about that other than to actively drive users away from the site. (Difficult captcha? Erlang Fridays? Comic Sans?)
The other part of the problem is that the karma system rewards commenting. It isn't considered appropriate to downvote a comment unless it is overtly offensive or incorrect, so mediocre commenters don't receive signals when they are contributing almost nothing to the conversation. In fact, a mediocre commenter will comment more, because more comments mean more chances for random karma. And others see the mediocre comments and reciprocate. There is no way to reward someone for not commenting, even when it improves the site.
A number of the solutions already mentioned would help decrease the number of comments. One additional one: Make more of the site's behavior conditional on a high karma/post ratio rather than a high karma.