Have you tried:
– deactivating all plugins to see if this resolves the problem? If this works, re-activate the plugins one by one until you find the problematic plugin(s).
– switching to the default theme to rule out any theme-related problems?
– resetting the plugins folder by FTP or phpMyAdmin? Sometimes, an apparently inactive plugin can still cause problems.
– reading http://wordpress.org/support/topic/279132?
That drag and drop function is Javascript so make sure that you have Javascript enabled. Also clear your browser cache.
If you have the WP Events Calendar plug in – get rid of it, it messes up widget drag & drop plus alot of other problems.