Hahn-Banach is fundamental as a mean to easily obtain existence of objects in functional analysis. Basically, it expresses that any problem of a certain type which has no "obvious obstructions" has a solutionssolution. Even in finite dimensions, it is at the heart of the powerful duality in convex optimization (or the properties of the Legrendre-Fenchel Legendre-Fenchel transformation).
Hahn-Banach is also equivalent to the lower semi-continuity in the weak topology of convex semi-continuous functions, which allows to obtain solutions of many variational problems via minimization, for instance when sublevels of the convex functional are weakly compact. On the other hand, you have to work harder (use other input, e.g. regularity theorems) to state anything above the mere existence of your solution.
