How do I enable Java Web Start for ASDM Launcher (Cisco ASA management) in Mountain Lion?
If this is indeed a bug with your application and Java 7 as Mark mentioned, you can re-enable the previously-Apple-provided Java 6 and Web-Start functionality by following the steps from HT5559. The steps are below, along with commands to undo these changes, if they do not help.

Open Terminal in an administrator account (Terminal is in /Applications/Utilities) and run this command: sudo mkdir -p /Library/Internet\ Plug-Ins/disabled to create a directory for "disabled" plugins that you don't want to run.

Run sudo mv /Library/Internet\ Plug-Ins/JavaAppletPlugin.plugin /Library/Internet\ Plug-Ins/disabled to move your current Java plugin to the new "disabled" folder.

Run 


  sudo ln -sf
  /System/Library/Java/Support/Deploy.bundle/Contents/Resources/JavaPlugin2_NPAPI.plugin
  /Library/Internet\ Plug-Ins/JavaAppletPlugin.plugin


to link Apple's Java plugin to make it your current JavaAppletPlugin.

Run  


 sudo ln -sf
 /System/Library/Frameworks/JavaVM.framework/Commands/javaws
 /usr/bin/javaws



to re-enable Java Web-Start.



To disable the above, follow the directions below:

Open Terminal in an administrator account and run:


 sudo ln -sf
 /System/Library/Frameworks/JavaVM.framework/Versions/Current/Commands/javaws
 /usr/bin/javaws



to disable Java 6 Web-Start and replace it with the newest Web-Start functionality.

and Re-install the newest Java Runtime Environment 7 from Oracle.