LaunchBar has a manual Clear History button, a checkbox that lets you control whether clipboard history should be preserved across restarts, and a time or quantity setting for how far back it should remember.
Personally, I want LaunchBar to retain clipboard history for as long as possible (1 week), and never clear it between restarts. If anyone has access to my Mac, the contents of the clipboard is the least of my security worries.
I’ve also used Keyboard Maestro’s clipboard history successfully; I may have switched to LaunchBar purely because I was editing a book about it at the time.