We'd need to have a look at some statistics to know. I was only able to dig up statistics for the United States: List of motor vehicle deaths in U.S. by year. This statistics doesn't record the total number of accidents, just the fatalities. Perhaps a bit surprisingly for someone, the car travel was a dangerous business in 1930's. read more
But to answer the direct question: given that traveling the same distance resulted in 15 times higher probability of death in 1930 than today, I’d say we can conclude that accidents were quite common - for those folks who actually traveled in a car back then. read more
As early as 1908, auto accidents in Detroit were recognized as a menacing problem: In two months that summer, 31 people were killed in car crashes and so many were injured it went unrecorded. Soon thousands of cars jammed Detroit streets, driven by inexperienced drivers. read more