Board Game Arena

I've noticed this sort of thing on occasion.

User makes a bug report. A significant period of time passes (a month or more). No developer comments upon the report, even if just to open the bug, ask followups, or similar. Sometimes, this even happens to bugs with a significant number of votes and game impact on a popular game.

What exactly is the correct response in this case?

The correct response is to wait.

All games on BGA are coded by volunteers. As far as I know, BGA has a grand total of 2 paid employees, een and sourisdudesert (the founders). And I'm not positive that for both of them this is their only job.

They tend to focus on the website itself, rather than the games. While they do offer incentives for fixing other issues (see the "Bounty Hunt" posts in the Developers forum), it is still volunteer work to fix game issues.