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?
In every context, a basis is essentially a (not the) smallest set you need to understand, in order to understand the whole object in question.
For example, a linear transformation is determined by its action on a basis. Similarly, a complete theory is determined by which elements of a (propositional) basis it contains.
The virtue of the word "smallest" in the first paragraph is that, roughly speaking, every behavior over a basis is possible. For example, if $\{b_i\}$ ($i\in I$) is a basis for $V$ and $w_i$ $(i\in I)$ are vectors in $W$, then there is a linear transformation $T$ sending $b_i$ to $w_i$ (and by the above paragraph, it's unique). Similarly, given a propositional basis $p_i$ $(i\in I$) and a map $t: I\rightarrow \{\top, \bot\}$, there is a complete theory $T$ such that $p_i\in T\iff t(i)=\top$ (and by the above, it's unique).
These two aspects of bases (determination and freedom) mean that they're invaluable for understanding the things that act (in a very loose sense) on them - linear transformations, or complete theories, or etc.