This isn't a real answer, just some stats that are too lengthy to fit into a comment:
Looking over the current front page, the vote score for a question actually looks like a pretty good heuristic for predicting the question's quality. That is, questions voted +2 or higher look pretty legitimate. And the ones voted -1 or lower look less ideal, in terms of being the type of questions we want to encourage on this site. This suggests that the voting members of the community are still successfully identifying good questions -- the problem we're perceiving is just the ratio of bad questions to good ones.
At the time of writing, out of 48 questions on our front page, 11 of them have a vote score of +2 or higher. 6 have a vote score of +4 or higher. 10 have a vote score of -1 or lower.
Comparing to other SE sites: On Stack Overflow, 9 of the 48 front page questions have a vote score of +2. 2 have a vote score of +4 or higher And 2 have a vote score of -1 or lower. (Note that Stack Overflow gets a much higher volume of questions; the vast majority of their questions have zero votes in total)
On Superuser, 12 out of their 48 front page questions have a vote score of +2 or higher, and 10 have a vote score of +4 or higher. 4 have a vote score of -1 or lower.
Per-site front page stats ("voted up" == +2 or higher, "voted down" == -1 or lower)
- Stack Overflow: 20% voted up, 4% voted down
- Superuser: 20% voted up, 8% voted down
- GameDev: 20% voted up, 20% voted down
It's interesting (and perhaps surprising) that the percentage of up-voted questions on the front page turns out to be almost identical between the three sites, even with Stack Overflow's substantially higher churn rate. The real difference between the questions shown on the front-page of the different sites seems to be in how many of the non-great questions are bad enough to attract down-votes.
I wonder whether any other SE sites have down-vote rates similar to ours? We might be able to learn from them, if they've dealt with this situation in the past.
For the sake of interest, here's a graph showing the percentage of GameDev.SE questions asked in different months that were downvoted to -1 or below:
Interestingly, this shows a big peak of downvoted questions a little over a year ago, after which it fell right off, only to have a few brief peaks later on, including a big one that appears to be going on right now, which lends some support to @Anko's suspicions. But even these big peaks are topping out only at about 15% of questions being downvoted.
But here's another interesting graph, this one showing the percentage of questions asked in GameDev.SE each month which would eventually wind up with a score of at least 4:
Note that this isn't a cumulative graph; pretty regularly, month-on-month, we've had a lower and lower percentage of questions reaching a +4 score ever since the site opened. In July of 2011, nearly 50% of all questions asked were reaching a score of at least +4. In July of this year, only 4% of questions reached +4. (To be fair, though, we got almost twice as many questions in July of this year as we did in July of 2011)
The query I used is saved publicly here. It prompts you to specify a minimum threshold, and graphs the percentage of questions which scored beneath the specified value.
Please feel free to mock my rudimentary SQL skills, and let me know about the terrible blunders I've undoubtedly made which completely invalidate these graphs.