I believe the driver entry is case sensitive. I tried editing mine from "alsa" to "ALSA" and subsequently winecfg complained that it was a non-existent driver.
If alsa isn't working for you, you can also try using the esd driver, since pulse was designed to be a drop-in replacement for it.
In addition to using regedit (HKEY_CURRENT_USER\Software\Wine\Drivers), you can also edit the ~/.wine/user.reg file directly. The sound driver entry is this:
Muliple drivers can be separated with commas ("Audio"="esd,alsa"), although as the dialog says, this isn't recommended. My wine shows alsa, esd, oss, jack, and nas as options.
BTW, the number string on the first line appears to be simply the time (in epoch form) when the setting was changed.
And just for completeness, the DirectSound settings at the bottom part of the dialog appear to use this entry:
DefaultBitsPerSample can be 8 or 16, and DefaultSampleRate has options for 8000 11025 16000 22050 44100 or 48000. 44100/16 is CD quality, of course.
EmulDriver is Y or N. I'm not 100% sure what this does, since there's no direct equivalent for it in the gui. Best to just leave it on, I think.
HardwareAcceleration can be either Full or Emulation.