It's quite an antique phrase used by certain of our enemies to insult the US together with our allies; e.g., "the Americans and their running dog allies" (the Brits, Israel, etc.). I seem to recall hearing it from the North Vietnamese.
I just found this from Wordsmith.org:
running dog (RUN-ing dog) nounA servile follower; lackey.
[From Chinese zougou, from zou (running) + gou (dog), apparently as an allusion to a dog running to follow his or her master's commands. This term was employed in Chinese Communist terminology to refer to someone who was considered subservient to counter-revolutionary interest.]