Is there anything that makes a 10 base system superior to an 8 or 12 base system?
What if the Aliens in question only had eight fingers? Would they come up with an 8 based system. 
(Okay, serious question but this last part was a joke)
Well, the base doesn’t really matter - the critical element is the concept of zero. It took humans quite a while to get it; I can picture an alien species somehow finding an alternative.
Why would an alien species have fingers? 
But as Bryan says, the base does not really matter. If the did base 2, they might figure out digital systems more easily…
Since humans on their own have come up with non-base 10 numbering systems I see no reason aliens would be particularly likely to settle on base 10.
Base 60 systems occurred in ancient Mesopotamia and gave us our 60 unit time divisions (seconds, minutes, hours) and 360 divisions of the circle.
The Mayans used a base 20 system.
Europeans used base 12 for some purposes, hence “dozen” and “gross”, even while mostly using a base 10 system.
That’s just a few examples.
I’m not sure why that last part is a joke. If they had 8 fingers, they would more likely use base 8, as Robot Arm’s documentary pointed out (how many digits on that robot arm, anyway?). I’d say that, if we were being logical, we’d choose base 8, 12, or 16 instead of base 10, since those numbers are much more divisible, and 8 and 16 are much easier to convert to and from binary, which is a computer’s natural base.
Indeed, no reason for even humans to settle with ten. The Babylonian method of counting to 12 in one hand, and then to 60:
You do it by counting your finger segments.I remember that it came by the Mayas realizing that they could count also their toes, 20 digits really, so it made sense to have a base 20.
So even if the aliens had 10 fingers or tentacles there is no rule that they would had arrived to a base 10 system.
Eight fingers total? The aliens may have to use a base-4 system if they were quadrupeds. They would lift on forepaw and count 1, 2, 3, 4 … but if they lifted their other forepaw they’d fall flat on their face and bloody their noses.
ETA: 60 is a nice number in that it’s evenly divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30.
If you have a system of writing, there is an advantage in larger bases. In base 10, nine is 9, in base two, 9 is 1001. I’d hate to write one million in base 2,
Only, as noted above, if you have a zero. Try writing in Roman Numerals, which are a representation of a base 10 system.
I know the Romans didn’t have a 0, but this confuses me. If you asked a roman child this question “Ceasar has 2 apples, Brutus took two apples from Ceasar, how many apples did Ceasar now have?” what would the roman child say?
The Romans would have had a word for ‘none’, I’m pretty sure. The real advantage of the Arabic zero is not being able to answer the question above, but in being able to make numbers like 10 and 20- thus, not only zero, but positional notation, in which the symbol 2 can mean two, twenty, two hundred, or two thousand, depending on the other digits around it. But you can’t get a system like that to work without a zero.
The Romans didn’t really have that. A V pretty much always meant five… maybe you could underline it to multiply it by a thousand or something like that, but there’s a limit to how many times you can underline XVII 
As has oft been pointed out, a base-ten system is not as handy as a base-twelve system. But most anything will work if you have a place-holder system.
Show me a picture of the aliens. I’ll give you some educated guesses on their number base. Although as noted above, it won’t necessarily be as simple as counting their finger-equivalents.
Would a sentient starfish count in base 5, or in base 4 since he’d be using one appendage to enumerate the others?
A federation of aliens, each with different bodily features, would probably come up with a base 8 or base 16 system, since they would certainly use that for their computers no matter how many digits they have. Base 10 is about the worst non-prime choice in the first 15. (Yes 15, I’m a world authority on base 1 arithmetic.)
It’s worth pointing out that however many digits these aliens use in their number system, they will call it “base 10”.
23, same as always.
Is that counting the toothpick?
I’m reading a sort of old-fashioned space opera series that deals with this. The valiant Earthlings fight alien races with different digits, and the story is told from both perspectives, so you get alien troop numbers in a base 8 system, then the humans note how they under attack by 64 fighters.
A base-12 system does seem somewhat superior to a base-10 system. In a base-12 system, it would be easier to identify multiples of two, three, four, and six because they would always have the same final digits. In a base-10 system, you only get this with multiples of two and five.
Well then, how many tentcles on each pod… maybe they have 4 tentacles on the right pod and 6 on the left… or even a variable number on each, then what base would the use?