I'm having exactly the same problem as the most recent tester on WineHQ's AppDB. That is, StarCraft.exe consumes one core of my CPU and nothing happens. I've saved a complete log of terminal output here.
Unlike that tester, who was able to run the game with wine-staging 4.8, when I try with winehq-staging 4.12 even the Battle.net launcher doesn't work properly (this was the subject of my previous question). That seemed to have something to do with Vulkan; the launcher does work with stable 4.0, but I still get the following errors when running vulkaninfo:
ERROR: [Loader Message] Code 0 : /usr/lib/i386-linux-gnu/libvulkan_radeon.so: wrong ELF class: ELFCLASS32 ERROR: [Loader Message] Code 0 : /usr/lib/i386-linux-gnu/libvulkan_intel.so: wrong ELF class: ELFCLASS32