There is no systematic and universal solution to this matter. Instead, there are myriad approaches, and individual justices adhere to whatever they hold to be correct from a general jurisprudential perspective. The question of the proper principles for interpreting legal language has resulted in the spilling of much ink, for example in the works of Mellinkoff and Scalia. There are a number of publications by Tara Smith which address the question from a philosophical perspective, where she outlines classes of theories, such as various kinds of "originalism", which includes "original intention" and "original meaning". Generally speaking, these are not theories of language and law, and are not linguistically-informed theories.
Usually, the question of meaning change does not arise. While word-meaning does change over a long period, there haven't been particularly significant changes in word meaning within the history of the US. A salient example (discussed by Smith) is whether the definition of "arms" (2nd Amendment) has changed in the past 250 years. Although the set of things that constitute "arms" has changed in that period, the definition of the word has not changed. However, meaning changes such as "against" or "meat" are bound to happen in the future, so I am not implying that this couldn't be an issue; it just isn't quite one, yet.
Statutes are frequently revised so that words whose meaning seems to have changed or at least are unclear get replaced. Constitutions are more resilient, and don't get re-written. There are some meaning-related issues in the Constitution, especially words which don't have clear meanings in contemporary ordinary English, such as "apportioned", "emolument", "impost", and outside of the Constitution there are terms like "fee simple" which are meaningless to most people. In legal circles, these terms are understood, because we have legal dictionaries. Similarly, "lay" is no longer used to mean "levy" (taxes), but it is still understood that that is what it means to "lay taxes".