Yeah, ideally we want our base to be the LCM of the largest possible set, while still remaining small enough to be usable. base-12 does a great job of this compared to decimal since it is the LCM of 1, 2, 3, 4, 6, and 12. 60 is a multiple of 12 and it's factors, but also of 5, 10, 15, 20, 30, and 60.
Binary's main utility is with electricity based computing, where there's an on and off state.. Same goes for base-16, it's only a multiple of 1, 2, 4, 8, and 16, but two digits are equivalent to 1 byte (8 binary bits or digits) of data, which makes it convenient for CS people.
MB
Some people like octal for CS stuff too, but I despise it since you can't split a byte into octal very easily.