I am coming from the other direction; I have ridden dirt bikes for many years and my 09 KLR is my first street bike. Also, both of my sons ride dirt bikes.
I like the KLR, but would not consider it for a moment for your purposes. Your son will inevitably want to explore- far and fast- and you will be miserable trying to follow. The KLR is too heavy and doesn't seem to want to do much more than fire roads.
When I made the decision to buy the KLR, I was trying to decide how much dirt and how much street I'd realistically be riding. I chose the KLR because I already have a dirt bike and would mostly be commuting and hitting local fire roads.
Had I opted for a dirt bias, I would have bought an XR- or for a serious steeet legal dirt bike, a plated KTM. I don't know which state you live in, but we have all kinds of registration problems with plating dirt bikes. It is possible that you could get aroiund this in your area. In that case, any trail bike like the Yamaha WR or Honda CRF-X would be great.
Hope this helps.