I am currently saving for my first telescope which will be a large aperture dob on a computerized mount. While I am doing that I thought it might be fun to pick up some relatively inexpensive binoculars to begin wandering around the night sky.
Of course the first thing I looked at were a big pair of Celestron 25x100 Skywatchers but after reading many tales of woe and a lot of bad reviews I am no longer considering those. Many seem to recommend something in the 10x50-12x50 range and I have been eyeing a pair of Nikon Action Extreme 12x50s on Amazon for a few days now.
One thing I should mention is that hand-held observing is not an option for me as I have "essential tremor" which is a condition that basically causes my hands to shake under any significant load. So I will absolutely have to use a tripod for any size binocular. This is the primary reason I'm considering the 12's over the 10's, not that I expect a lot of difference.
For what it's worth, there is quite a significant difference between 10x50s and 12x50s; magnification generally matters more than aperture for astronomical binoculars. But stepping back from that specific question, you're facing a basic tradeoff.
The higher the magnification, the more faint objects you will be able to see, and the more detail you will see in bright objects. The difference between (for instance) 10x50 and 15x70 binoculars is truly dramatic. Take galaxies as one example. With the 10x50s, you're basically just detecting galaxies; you won't see any detail whatsoever except in M31 and M33 -- and not much even in those. With the 15x70s, a number of other galaxies begin to come to life, most notably M82, but also others. With 10x50s, you can detect all the Messier objects under dark skies, but some are quite difficult. Under identical conditions, all the Messier objects pop out instantly through 15x70s.
However, the whole point of binoculars (for me, anyway) is their huge field of view. And the higher the magnification, the smaller the field of view. So if you have no familiarity whatsoever with the night sky, you might well find the extra field of view of 10x binoculars -- or even 8x, 7x, or 6x -- outweighs the hugely improved light grasp and resolution of the 15x70s.
Ideally, you would own at least two different pairs of binoculars. But that's another problem for another day; your immediate problem is which one to start with. On which subject I have said all I have to say.
If not for your tremors, I would say that the hand-holdability of 10x binoculars tips the balance solidly in their favor. But since you will be using a tripod in any case, it's a toss-up. Only you can decide.