It’s not a burning issue, but while watching a show on TV the other night, it occurred to me that the earth is a bit unique in that it has only one moon.
Say that we had two moons. How would NASA have handled the moon landings? Would they have just picked one moon and gone there the whole time? Or would they have tried to land on both? Or the closest? Or the biggest?
“We choose to go to the moon. We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one which we intend to win, and the others, too, and also because if we get it wrong, we can try to go to the other moon.”
How big would these two moons be? If one were significantly larger than the other, we’d have gone to that first, because we’d have pointing to this obvious world in the sky since prehistoric times. Mars’ moons, for instance, just look like points of light from the surface, and there might not have been a big project to get to either one of those. On Mars, the ‘ancients’ might not have realized much difference between them and planets.
As much as it pains me to fight the hypothetical, I don’t think there is much you can say about what we would have done. You might as well ask whether we still would have went to the moon had the Roman Empire never existed.
The presence and peculiarities of our single moon have influenced humanity far more than you would expect. Without substantial tides, there is a chance land-based life might never have evolved at all. Perhaps, with two moons, or no moon, we would have discovered heliocentricism earlier, or maybe never. The singular image of our large, bright moon has probably influenced culture and society more than any other astronomical body except for the sun and Earth itself. We may never have been driven to explore space without it.
Isaac Asimov wrote a couple of interesting essays on the topic: The Tragedy of the Moon & The Triumph of the Moon.
…we’d have one satellite, and one satel-heavy.
It would have added a wrinkle to the space race. “Okay Leonid, you send your Cosmonauts to this moon, and we’ll send our boys to that moon, and we’ll just see who gets to their moon first.”
We’d have gone to the one we thought would be easier to get to - unless we thought the Russkies were headed to the other one. We needed to get our flag planted first!
On Soviet moon…
…
…
…anyone want to give it a shot?
… green cheese eats you.
… blue cheese moulds you! 
Dark side faces you 
No, no, red cheese, obviously.
… tranquility sees you.
(bit of a stretch, I know)
I look at Mars’s situation, and it makes me perfectly happy with just the one moon we have. What a couple piles of crap Mars’s two moons are.
Not just that, but we’re AMERICA (FUCK YEAH!). I doubt it would occur to American rocket scientists to ever go to the smaller one. Maybe to Nazi or Commie rocket scientists, but not to Americans.
Not until somebody thought it might be full of gold. THEN we’d be all over it, Moon Maidens be damned.
“I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the Moon, whichever’s closest, and returning him safely to the Earth.”
The only thing we can state with complete assurance is that it would have cost more to go to one of two moons, than it did to go to the one moon we have.
The cost of feasibility studies, and committee meetings alone would have been an extra few billion.
Tris
Sorry, but JFK was Navy. The correct quote would be, “I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the Moon, whichever’s BIGGEST, and returning him safely to the Earth.” Little Moon was not worth mentioning, much less aiming for. 
I think we would have had to declare one of them an enemy moon.