Just like you would any other other plant. Now is a good time it is dormant. It might be of help to identify where you live and if it is snowing. Is it from Madagascar? Where does it come from? Species name would give us clue on how it grows. Most Aloe need heat but not all of them. Norma

I live in eastern Indiana (lots of snow outside). The plant has been indoors all its life. I was told that the aloe is 23+ years old, and I have no ID information on it. It has been growing on the sill of a south-facing curtain wall right above a heat register. When I repot it, would it be wise to bury the extended stem (approx. 16" inches from top of roots to first leaf)?

If you choose to repot this with a shorter stem, you could probably cut the plant in half, strip off leaves to leave bare stem for at least 4 inches, let it air dry for a week before repotting it, and then repot the two plants into individual pots with fresh soil. You can do this after it finishes blooming if you don't want to risk damaging the flowers, but I personally have cut the stems and left the plant without any roots and/or soil on several Aloes such as A. maculata and A. rubroviolacea, A. striatula while already budded up for bloom, without any damage to the flowering spike. Usually there are enough stored reserves in the offset piece to keep it happy until it is replanted and forming new roots, as long as it isn't shocked by staying too wet or getting too cold.

Your plant looks like it grows the way it does because it wasn't getting quite enough light in winter to grow more compactly, although I don't know if the species you have tends to send up multiple stalks if given more light; it looks as if it would/could if grown outdoors with natural light.