It seems like in most older works of animation, the majority -- if not all -- of the private cars on the road do not have a roof to speak of. Or, if the car is a convertible, the top is pretty much always down.
This is more than likely a device to allow people to drive by their friends on the curb and wave/chat unhindered. It also allows the artist to avoid the nuisance of drawing a window or shadows, and just draw the characters normally.
Although '50s movies like Grease
have open cars more often than not, this troper is not old enough to know if that was common enough at the time to be Truth in Television. According to The Other Wiki, convertible sales plummeted by the 1970s due to concerns about rollovers. They've made somewhat of a comeback since a form of roll bar was worked in, but they're still much rarer than certain media would have you believe.
Exceptions include: private limousines (though sometimes only the passenger is enclosed while the driver is exposed to the elements), police cruisers, settings in inclement weather, mobster "take him for a ride" situations, "locked out" situations, etc.
-- In Archie Comics, Archie's "Ol' Bess" jalopy has a top, but it's almost never raised, even in bad weather.
-- The "Flintmobile" in The Flintstones does have an animal hide roof (albeit one apparently flimsy and expendable enough for Dino to stick his head through it at the drive-in), and a rear hide with a gaping hole in it, there are no sides to speak of. Some episodes have no roof at all on the Flintmobile, but Barney's car, essentially a wooden log "pencil sharpened" at both ends, is an open vehicle all the way.