Often in America there is a perception of France as cowardly or weak due to their defeat in WWII. Has this idea always been prevalent since WWII? Did more recent events like France leaving NATO in 1966 influence it?

I think an important thing missing from the conversation here is that the French, during the Second World War, were very much so opportunists. Almost immediately after the Germans launched their Western offensive in May 1940 it became clear that the French were not going to be able to win.

Rapidly ensues a debate on how to best react to the French defeat, on the one side you have the likes of the then Prime Minister Paul Reynaud, who suggested that France should relocate its government outside of metropolitan France and continue to use it's colonial resources to fight the Germans. While on the other, you had further right like Petain and CIC Weygand (Recalled from Syria to replace the dismissed Gamelin), insisting that it was the duty of the government to suffer alongside the governed.

Now one of the interesting things that takes place in late May is Weygand orders the cancellation of a counter-flank offensive near Ardennes, then re-orders the same maneuver two days later having by then already failed, and then claiming that he "was recalled two weeks too late to stop the advance" A lot of speculation can take place as to whether or not those who were further right in the French political scene at the time actively sabotaged the efforts to counter the German advance, but it is really up in the air.

After the astounding defeat a lot of those on the right felt so-to-say disillusioned by the Republic and began to take a "woe is me" stance on the military defeat laying the blame for the defeat primarily at the feet of the problems that they saw with France prior to the war. Many also suggested that the opportunity could be used to fix these problems, see Charles Maurras, primarily that of "French Decadence" in the 1930's as noted by a wide range of authors during the period.

The entirety of French policy during the war can be seen as a how can we save our own a@#es doctrine, and as soon as the war ends, the French play the card of having fought with the Allies the entire war, when the reality is that had the war tipped the other way the would've likely had the same argument for the Axis victors. I think it is this conniving opportunism of the French during the war that gives them the coward characterization in contrast to the other players in the war.

Whether or not that characterization is fair or unfair is for each individual to decide.

Edit: 2 great books on the subject of France during the war are The Fall of France: The Nazi Invasion of 1940 and France: The Dark Years 1940-1944 both by Julian Jackson