Actually the idea of a 'base 10' number system is somewhat modern. I forget when it came into practice, but I know there were ancient cultures that used different numbering systems.
Actually the idea of a Base system is what really drove the science of mathematics forward. Other number systems were more of tally notes, based on addition. A Base N system centers around multiplication followed by addition, which allows the mathematical system to easily handle big numbers.
Think about it, in an ancient time how often does your average person need to count to more than 100 or so? Up until about the 1800s or so, few people really needed to deal with numbers, and for a big chunk of the world a lot of people were barely taught basic math skills till after 1900, and even then it isn't till something like the 1950s or so that the majority of people are expected to know.
But when you think of how we actually write numbers, and the theory of Zero, a base 11 or base 6 system actually makes more sense. Write it out while you count on your fingers. Show yourself Zero. Now start counting digits till you get back to zero again. Surprise,... Our hands aren't base 10, they're base 11.
If you are counting things out and writing them down, it actually makes more sense to learn in base 6, you hold your writing tool in one hand, and count with your free hand.
That humans use a base 10 system most likely does stem from our having digits on our hands, but this is most likely because it is a translation from a tally number system, not a pure mathematical 'base' system.
We would have learned to count before we made a numbering system, that is we learned one, two, three, four, five, etc, and THEN we learned to write 1, 2, 3, 4, 5, etc.