Stenzek rewrote the Windows BBA code in
PR 3432, which was merged in
4.0-9017. The old BBA code was far more prone to crashing, and I would crash at least once a week just from loading quests in the course of standard online play. The new BBA code included in every Dolphin release since then is far more stable than the old BBA code, but it can still crash, particularly if you load large quests frequently, e.g. Seat of the Heart, which is by far the most crash-prone quest. I also crash occasionally when joining a team hosted by a modem user, presumably because the time it takes a modem to transmit the join packet reveals some bug in Dolphin that doesn't occur when the join packet is sent instantly, as is the case when joining a team hosted by a BBA user.
When you join a team hosted by a BBA user, you'll see the saving icon appear twice on a black screen, and then you'll appear in the guild. If you join a team hosted by a modem user (or someone who's just laggy), you'll see the first save icon, and then you'll get a camera view overlooking Pioneer 2. Most of the time the camera will suddenly move to the guild, the game will save again, and you'll be on your way. Sometimes, however, Dolphin will just crash while you're staring at that overhead view, and this is particularly dangerous, because Dolphin doesn't write data to the memory card file at the exact moment you see the saving icon. As such, if Dolphin crashes while it's writing to the memory card file, your memory card corrupts, and you lose all four charactes on it. As such, be sure to make regular memory card backups if you play online and not just weapon backups. Even so, one crash per month is a lot better than one crash per week (or several crashes in a single day of quest testing).
Sadly, PSO isn't in the top tier of popular GC games, and the BBA isn't an important peripheral, due to the exceptionally small number of games that can use it, so there isn't a whole lot of developer interest in working on BBA-related bugs. Even so, I keep an eye on the recent development versions to see if any PRs of interest have been merged.
I don't think I've ever seen the bug you describe, but have you tried the other backends? D3D used to have a major bug with a giant white box covering the entire screen, and while that was fixed some time ago, perhaps you've discovered some remnant of it. I don't encounter graphical problems anymore when using OGL, thanks to all of the improvements since 4.0.