I think the reason we’ve accidentally fallen on 10, aside from the fact that it’s the number of fingers we have, is the size. Any base we choose is going to be a balancing problem between the set of symbols and the length, in symbols, of the representation of common numbers. The obvious examples would be that base 2 is clearly too small because even a common number like 20 requires 5 digits, where a base like 60 requires memorizing and recalling that many different symbols. Now, I don’t think the number of symbols is strictly limiting because most languages have alphabets at least 2-3 times larger than the number symbol set. As others mentioned, the real problem is the limiting factor of being able to memorize and quickly recall simple arithmetic which gets harder at the square of size of the symbol set. So in that sense, I think 10 is a pretty good compromise on the set of symbols. So I’d say chances are any base chosen ought to be relatively close to 10.
The part where I think base ten fails is, as others mentioned, is that it’s not divisible by a lot of common fractions. Unlike SCSimmons, I don’t think limiting to just primes is a necessary constraint because even though 4 isn’t a prime, it’s probably more useful than 5 for most cases, and certainly more useful than 7. I saw a method somewhere where the author was arguing for a base 12 that related the size of the base to the number of divisors, so, for instance, base 10 gets a ratio 5, where base 8 and base 12 are both relatively close in size to base 10 but have better ratios (4 and 3 respectively), and are divisble by more useful values (4 alone is more useful than 5, 12 has the added bonus of being divisble by 3 as well). The next closest numbers to 10 that have a better ratio are 6 and 24, both of which probably fail the size problem. So based on this, I would argue that base 8 and base 12 are probably the two best candidates for bases.
Finally, with the prevalence of computers, being a direct power of 2 is probably somewhat of an advantage, but probably a very small one since dealing with base 10 isn’t a problem with modern computers. I don’t think it’s enough to overtake the better ratio and thirdability (is that a word?) of 12, but it’s still a slight nod in it’s favor.
So if I were to choose an optimal base, I’d go with base 12 followed by base 8 not too far behind. I’d say base 10 is probably third. Finally, I think base 16 and base 6 are roughly tied for 4th. Base 16 suffers from being a little large and having a slightly worse ratio than 10; however, it does have the benefits of having a better set of divisors (4 is strictly better than 5, IMO) and being a power of 2. Base 6 has a good ratio and is thirdable, but it’s almost certainly too small and strictly inferior to at least 12 (same ratio, divisible by 4, better size).
So I’d say we didn’t do too bad with choosing 10, but probably could have done a little better.