Right now, we have about 12 questions at -3 or lower on the homepage. This is after I closed and mod-deleted questions that are explicitly off-topic and very, very unlikely to be salvaged. It seems like the threshold for hiding questions from the homepage may be -4 or -5.
I propose two things:
For logged-in users, hide questions that are -3 or lower from the homepage. Hopefully, at least one person has also cast a vote to close if it should be closed and kicked it to the queue. If it's not already the case, I would suggest that at -3 a post gets put into the Low Quality queue for review, even if it's not on the homepage.
For logged-out users (and perhaps logged-in users under a certain reputation threshold), hid questions that are -2 or lower from the homepage. It would be nice to prevent visitors (who, hopefully, register and become members) not get the wrong idea of our community. Voting (and the rationale behind it) may not be obvious. I'd rather err on the side of caution and hide posts that are likely to not be a good fit.
I'd also encourage some network-wide data analysis to check on thresholds for this across the network. Perhaps the current thresholds require too much voting on other sites so there may be opportunities to propose changes to those communities or make default configuration changes.