There are a few important things to note, in addition to all the things mentioned.
As said, 10 isn't special, other than humans having 10 fingers.
In addition, 60 isn't special, either.
360 is special, but only to humans, because that's really close to the number of days in a year. Early calendars were based on 360, and (some) early number systems co-evolved with their calendars. (If you've ever played with the GRAD setting on a scientific calculator, having 400 GRADs in a circle would make much more sense than the 360 we ended up with.) The Sumerians liked 60 because it divides so nicely into 360, and 12 is nice because we have 5 fingers on our hand, and 5 12s gets us back to 60.
Why are there 7 days in a week? Because 7 divides so cleanly into the 28 days in a "month" - i.e. the number of days in our lunar cycle.
I did a research project on the evolution of number systems as part of my undergrad work. From 1 to 0 was the seminal work on the topic. Number words and number symbols was also a big chunk of my research.
An alien species (assuming, like us, it couldn't divest itself of what it's been used to for millenia of growth) will base their numbers on whatever is relevant to them. Number of appendages is important, but so is working with constants given to you by your environment. You may get a really interesting system by there being multiple moons or stars in their system.
One last interesting note - humans and other animals have an ability to easily recognize groups of up to 4. After that, it becomes increasingly difficult. Even if we had 8 fingers per hand, it's possible tally marks would be broken up in 4s or 5s, just because it's easiest to read them that way.
I feel a need to add a bit more; the comments to my answer are arguing over minutiae, and it's obscuring the important point.
The number system of your society will form extremely early on - it will coincide with the birth of your society and the formation of their earliest languages. To know what your alien's number system will be, you have to think of how their society was first created, and what phenomena they would observe while doing so.
Number words start with the simple, familial concept: "There is me (1), us (a few), and others (many)."
The next immediate growth is grouping: "I thought there were more of my family. Is someone missing? How many are we?" At this point, you start grouping using the number of some appendage. For humans, the easiest thing is the five fingers on your hand. They're always around, and you can pick up and put down each finger as needed: "There's me. There's Ugg. There's Ock. There's Uga. There's Gug. That's one hand. And there's Gaa. That's one-hand, one-finger. We were one-hand, two-finger yesterday. Where's Kaa? WHERE'S KAA????"
Number concepts don't evolve beyond this much (humans went from one hand to two, then stayed there) until you go from familial to tribal, and have enough safety for there to be someone who focuses on observing. This would be a priest, or a medical man. They'll notice something that recurs on a predictable scale. It has to be a predictable scale, really, because there is not sufficient level of intellectual sophistication to find a non-regular repeating pattern. At this point of time, the scientific discovery is that the pattern exists. That "ability to predict the future" is part of what will make this observer powerful. Whether (s)he will "make the sun/moon come back" or will "make water erupt from the ground" (a la Old Faithful), their knowledge will make them powerful, so they'll develop it. Whatever number this thing recurs at will likely become the society's radix. It's likely that this counting will be of something astronomical - either day cycles or moon cycles - because there aren't much more precise measurements. Hours won't exist for tens of thousands of years yet.
This is what you're contending with when you look at a number system - concepts that go back as far as society itself, at the formation of a species' first words.
It's important to realize that numbers weren't chosen by the society. They found the numbers based on themselves and their environment that best helped them to survive, and their culture, math, and science all grew up based on these values. The cultural inertia of such numbers is really hard to overcome.
Number words - and the numerical systems that derive from them - must have their origins in observable phenomena. At the time that the number words are formed, there just isn't enough strength in the culture for them to form any other way. There will be little decision-making in this process - it would have to be an easy enough decision to make that society would actively make it over tens of thousands of years of the most primitive culture possible.
A spacefaring society will have developed their mathematics enough to realize there are other, likely more efficient bases. Their specialists will use them in their specialist tasks (as we do for computers, and for speaking in abstract number theory about better bases.) Particularly, their rocket scientists could use number constructs that most others do not. But the people will still use whatever number system their society grew around, based on their physiology and their local astronomy.