In this century, we certainly can: do not have license plates at all, use radio beacons instead, which send encrypted copies of the plate. The police would be allowed to request decryption from the DMV (e.g., "running the plates"), and a few other authorized people, but the general public could not. The secret keys would be the weak point, and would have to be held only in a limited number of DMV data centers (which would help in establishing an audit trail when plates are accessed).
I'm not saying it is necessarily something that should be done, but it certainly can be done.
Sure, but then how do you tell the police that you just saw a black SUV with Massachusetts plates ending is 907 blow through a red light and almost run down an old lady with a cane at 7th and main? How would they ever find the driver or owner?
While there are many ways to abuse the fact that license plates are publicly-visible (the aforementioned abortion protester being one example), there are still very legitimate reasons they should continue to be that way (holding people responsible for their actions when they infringe on the rights of others, as in my example). This isn't to say that we should always and forever attach a stamped piece of metal to cars, but the current system does have the advantage of being compatible with the Mk. I Eyeball.
While we're on the subject of a slippery slope of increasing police power, we might as well assume that eventually the people will have zero power and zero responsibility with regard to crime. In this hypothetical future, you won't need to tell the police you saw a black SUV almost run down a pedestrian; it's a police problem. This is the illogical end of abdicating personal responsibility in favor of total surveillance.
Maybe I am naive but it seems like there would be a pretty small sub-set of the population that could actually do anything with someone's plate number that would be an abuse of the system. There isn't some publicly available API into the DMV records is there?
A license plate is already essentially encrypted. It's a random sequence of letters and numbers that correspond with a unique individual's information. It contains no personal information in itself.
Except that your car shows everyone the same license plate every time they look at it. This would be completely unacceptable for a cryptographic solution, which would use fresh randomness to respond to every request (in theory).
I'm not saying it is necessarily something that should be done, but it certainly can be done.