If you want the distance between the mirrors in the two arms to be identical, you can tune it - but as is noted in Bas Swinckels' answer, they don't need to be so close together to do their job; and in fact, they are not tuned to be exactly the same length.
However, I would like to explain how, in general, if you have a large "Michelson interferometer" type of setup, you can get these distances ridiculously close.
If you have no idea how far apart your mirrors are, you can start by sending a short laser pulse down each arm and measure the round trip time; assuming you time to about 10 ps (which is really quite easy), you now have the distance to 1.5 mm (round trip distance uncertainty 3 mm).
At this point, you can use a narrow band light source to look for interference fringes: as long as the coherence length of the light is greater than the difference in path lengths, you will be able to see some fringes; as you move towards the position of zero path difference, the fringes get brighter. Depending on the width of the spectral line you use, this can get you quite close - say within 5 wavelengths of the "zero path difference".
Once you get really close, you can use a white light source to repeat the experiment: this has a very short coherence length, so while you will get constructive interference at zero path length with a near-black fringe on either side, you will lose the interference pattern very quickly. This can be used to find the "absolute zero path difference" distance - where the white fringe is centered in your field. For the LIGO experiment, you actually need a dark field (greater phase sensitivity), so you need to be off by half a wavelength.
An example of what I'm talking about is show on this site:
Once you have the zero path difference, you change to your very stable lasers, and tune the distance one last time to get the most perfect extinction you can. Because when you are looking for such a tiny phase shift, you will only detect it if you're working close to zero (so that the measured intensity is proportional to the phase difference).
Note that with a sufficiently stable laser, you might think you could be operating several wavelengths away from the "zero"; but when the phase differences you are looking for are this small, you really can't afford to allow additional phase jitter from any source to creep in. And the only way to prevent that is to make sure you have the same number of wavelengths in both arms.
And then you wait...