So you've tried to install this and found that after installing and allowing you to authenticate it will suddenly crash. Luckily there is a fix and it's quite simple.
Open the Crossover Bottle Manager and select the bottle in which you've installed the Battle.net Desktop App. Now choose the Control Panel tab, select the Wine Configuration item from the list and launch it.
In the window that pops up, choose Libraries tab and in the box that says New override for library enter dbghelp:
It will probably show a popup dialogue box like this one, ignore it and click Yes to proceed:
In the window that appears choose the Disable option and click on OK, click Apply and then OK to exit.
Try to run the Battle.net Desktop App again. It should run.
PS: I've created a crosstie that should handle all this for you guys automatically. It's currently waiting for ninja approval online.