Mail.app used to have the ability to hide the unread count badge, as shown on this 37signals blog post:
It seems that Apple has removed this in OS X 10.8 Mountain Lion. Is there any way to restore this functionality and remove the badge?
With the advent of Notification Center, the display of badges on any app can now be managed alongside display of banners, popups, and other notification-related settings.
Go to the
Notifications preference pane in
System Preferences, and select the Mail app from the list - the option to turn off the Dock badge should be there.
You may need to restart Mail for this change to take effect, though on 10.8.4 I do not.