Prediction: your prediction is wrong.
We pretty much all drive cars, how many of us are mechanics?
More importantly, the amount of under-the-hood knowledge to operate a car was diminishing, not increasing over time, and I expect the same thing to happen with computers. They will be all around us (already are), but for most people there will be no need to acknowledge their presence, let alone know how to program them.
Cars are not a good analogy for this situation, at all. The presence of software and computers in our life is much closer to the invention of the alphabet and writing. Written text gradually becomes so ubiquitous in life that being illiterate is a serious handicap for anything you want to do.
Likewise, the amount of software and technology we deal with on a daily basis is increasing at an exponential rate. It is just a matter of time before the ability to deal with it at a deep level becomes a necessity of life.
Of course, not everyone will be paid to write code, just like not everyone is paid to read/write, but you'll be handicapped in life if you don't know how to do it. Moreover, coding itself will largely change as well: programming languages are constantly adopting every higher levels of abstraction and gradually becoming more accessible to a wider audience.
