In a few situations, I found myself being asked by younger students why the concepts of a basis was important.

First, in the concept of linear spaces, it's easy to explain that having a basis allows us to describe the spaces with a relatively small set and makes working with linear transformation a much easier task (there are many more reasons, but this was an introductory linear algebra course).

When I was asked the same question in the context of propositional logic (a basis here is a maximally independent set of formulas), I found it a bit more difficult to motivate the definition.

In general, when asked "What is the importance of a basis?", what do you reply?