Please follow these troubleshooting steps:
1) Please make sure that you use "Run as Administrator" to install and to start the game (right-click on the shortcut or setup file and select "Run as Administrator").

2) Please reinstall the game AND add it to your anti-virus/firewall software's exception/trusted list. Simply disabling firewall or AV software could not be enough, since some of these software are running still in background or in services list. If you are using Galaxy, then please add Galaxy as well.

3) Reinstall the June 2010 DirectX runtime package. You can find the archive file containing the 2010 package installation files here:
http://www.microsoft.com/en-us/download/details.aspx?id=8109

You will have to extract the files into some folder, then open that folder and launch the DXSETUP.exe file in order to proceed with the installation.

After that please download and install DirectX Web Installer:
https://www.microsoft.com/en-us/download/details.aspx?displaylang=en&id=35

Please have in mind that this will only add DirectX 9.0c to available pool of DirectX libraries, and it will not overwrite your other versions. Some games are not able to utilize newer DirectX libraries, and they require DirectX 9.0c.

4) Please copy the contents of the Desperados.zip archive from here:
https://dl.dropboxusercontent.com/u/106372714/Desperados.zip
to the game's "Game" directory (default: C:\GOG Games\Desperados Wanted Dead or Alive\Game) and try launching the game. There might be some minor glitches in the main menu, but the rest should work ok.