Found a solution! See this blog post.
When I tried to start the game today, it told me that 1.0.3 is available and that it needs an upgrade. After 49% of the download, the BlizzardDownloader.exe crashed. So I started it again. And again. And again.
The problem is always the same:
wine: Unhandled page fault on read access to 0x00000000 at address 0x245b44c (thread 0019), starting debugger...
I’ve opened a support ticket on codeweavers.
According to some comments in Wine’s AppDB by MrNice, restarting the installer eventually solves the problem.
[Update] It seems to help to delete the sc2-* files in the program directory and the file
sc2-22660-22801-x86-Win-enGB-bnet-bin in the Updates folder. When I restart the game, I get the error dialog. Sometimes, a bit of the file has already been downloaded, sometimes, not even the changelog has been loaded. After closing the error dialog, the download resumes and hangs again at 49% with 3.8MB left.
[Update 2] Some people say that you just need to try often enough. That didn’t work for me. But I found a couple of *.torrent files in the installation folder which I showed to Vuze. After a couple of minutes, Vuze really started to download them … well, 4 of 5 of them. So I can’t play because no one has the 92666 byte file “sc2-16223-16291-x86-Win-deDE-locale”. Tragic.