I don't have a car, though if I had a car of the same vintage of my aircraft I probably would. I have a friend with an open-top Lancia from the 1920s and when he gave me a lift, helmets were provided.
It's interesting to look at how cars have become heavier as safety standards have improved. The weight of an original mini was 590kg; the new version is 1150. Much but obviously not all of the increase in weight will be down to improved passive safety features. Even the new 2-place smart car weighs more than the original mini.
I would consider not wearing a helmet if I had an aircraft with a properly designed cabin and airbags, such as a newer SR22. But realistically aircraft will never have the same level of passive safety as cars - they would just be too heavy.