Now, about your car being new vs old, that part is usually covered by "Comprehensive" insurance. It applies only to your car. If you think your car is too old and of not much value, don't buy comprehensive. It is optional unless you have leased/financed the car. If you own the car outright, then you can get rid of comprehensive if you still have it on your policy. Note: I don't know if all states have same laws but generally, this is the case.
The part of insurance where you usually don't want to cut corners is the liability insurance. That is, what happens if someone else's car or life/health is damaged by you.
There's possibly also a correlation between new cars and accidents (for the same reason sports cars cost more), though this may be somewhat offset by the probability that injuries probably go down as a function of recency of safety technology.
TL;DR - there are a lot of factors in costing insurance, but at least one of them should go down in direct proportion to time.