Wow, I’m pretty blown away that all the other answers are wrong.

The answer is complicated, so first let’s correct the assumptions of your question and put them in context.

First, it should be understood that in highly coherent monochromatic light, (single frequency) one can split a beam and then upon recombining them, aligning the same original peak or trough with its original partner isn’t necessary to create an interference pattern. They can be drastically misaligned and therefore it’s possible to get an interference pattern in a monochromatic light based interferometer experiment even when the distance between the arms is drastically different. No particular alignment is needed in highly coherent monochromatic light to create an interference or “fringe” pattern since it will simply alternate between the fringes of the Moiré pattern produced. This situation however, is highly undesirable for this experiment for complex reasons I will not attempt to explain in detail here. Suffice it to say it pollutes the results of the intended experiment and Michelson made sure to avoid that situation with a great deal of extra work.

Furthermore, beyond the ability for the lengths of the sides of the interferometer to be unequal, there is a little known effect (outside optics engineering) in monochromatic light called “lock-in” which causes any minor shifting of fringes to be ablated or nullified. Eliminating this effect in lasers is a required aspect of modern laser gyroscope designs where they must introduce multiple frequencies or noise to the signal to overcome the effect. (the reason why Michelson used white light when monochromatic light was easier to use)

So, starting all the way back with Michelson, he knew to avoid monochromatic light as an invalid way of performing this experiment. (incidentally, all classroom versions and replications of the Michelson-Morley experiment other than by Dayton Miller ignore these problems, use monochromatic light, and are therefore utterly invalid experiments)

So with pure monochromatic light, one can have a functionally unlimited amount of Fringe shifting caused by the shift of the phases between the beams and radically different path lengths on the interferometer without any problems in the fringe pattern. It will look apparently perfect, while hiding deep design issues.

Therefore Michelson used white light! So, what ingenious method did he use for perfect alignment?

So, with white light, you are almost correct in thinking that too much of a shift becomes a serious problem, but not as quickly as you thought. After a few wavelengths of yellow light, the different frequencies begin to spread away from one another. Notice the difference demonstrated below.

The slow spreading you see is what happens in the alignment of different frequencies. You can find more visualizations of that spreading process of multiple frequencies in Shiva Meucci's answer to Could a hologram from the holographic principle bend spacetime?

So what Michelson did is he used sodium light for alignment. It has a characteristic double fringe pattern (also known as sodium “d lines”) it creates in these experiments that is easier to find by eye when adjusting the interferometer. The process could sometimes take hours to complete even after Dayton Miller refined the process in later years.

Here is an example of the sodium double fringe:

And you can find more info on experiments using the periodic wash out of the double fringes to their advantage here: Experiment of The Month

We can find this process of alignment detailed in the original Michelson 1887 publication. Just remember that Michelson went through a lot of trouble to make sure not to fall prey to the problems of Monochromatic light which is now used in all supposed “replications” of the experiment. (IE they are all invalid)

That’s a little piece of easily verified history that is always absent from your history books. Unfortunately it is not just white light that is crucial, but also light travelling in a gaseous medium that is critical to the non-null experiments of Dayton Miller that won him the 1925 Newcomb Cleveland prize for proving the existence of aether, but that’s a controversial digression for another time.

Excerpt from On the Relative Motion of the Earth and the Luminiferous Ether (the original 1887 Michelson-Morley)

The paths being now approximately equal, the two images of the source of light or of some well-defined object placed in front of the condensing lens, were made to coincide, the telescope was now adjusted for distinct vision of the expected interference bands, and sodium light was substituted for white light, when the interference bands appeared. These were now made as clear as possible by adjusting the mirror e/, then white light was restored, the screw altering the length of path was very slowly moved (one turn of a screw of one hundred threads to the inch altering the path nearly 1000 wavelengths) till the colored interference fringes reappeared in white light. These were now given a convenient width and position, and the apparatus was ready for observation.