Good suggestion. I'm a cyclist and sometimes I walk around wearing my bike helmet. Some people give me funny looks, but I don't care. To regularly wear helmets, you have to recognize that avoiding TBI is more important than looking slightly silly or messing up your hair. Unfortunately most people don't get this. I think if helmets become more popular people will find that they don't actually mind the look, as you suggested in another comment.

Risk compensation is one potential problem with wearing a helmet. An example was mentioned in the original post: drivers passing cyclists who wear helmets closer. Also concerning is that cyclists ride more dangerously if they wear helmets (here's an article which I haven't read in full, but it discusses this issue). I imagine seat belts have a similar effect, as might drivers wearing helmets. Based on this idea backwards, I've read a proposal to add a spike to steering wheels to reduce dangerous driving.

Also worth noting is that avoiding long commutes and drives improves safety. Risk (probably) increases monotonically with duration.

One last suggestion would be to drive with a dashcam to make yourself slightly safer. I ride my bike with a very obvious helmet cam and I have observed that some drivers seem to drive safer around me when they notice it. I'm recording them, so if they make a bad move they'll be much worse off in court. I also think the camera makes me ride safer for the same reasons.