As a counterpoint to some of the other answers, actually Banach (and Hilbert) manifolds aren't quite as useful as might be hoped. Two indicators of this are:
All Hilbert manifolds are diffeomorphic to an open subset of the model space. (I think this also holds for Banach manifolds but am not sufficiently sure.) In particular, any Hilbert manifold is parallelisable so the tangent space contains no information.
If a Banach Lie group acts faithfully on a finite dimensional manifold then the group is finite dimensional. So there is no Banach model of diffeomorphisms (as a Lie group).
So Banach and Hilbert manifolds tend to be used as places to put other things. They are big enough to contain just about everything but simple enough that they don't add any extra complications. This leads to their use as extreme examples of "very big spaces" and means that for any particular situation one can often chop the infinite down to the finite (but very big). So, for example, with regard to representing K-theory, for any particular finite dimensional manifold there's a finite dimensional Grassmannian that will do but if you want to represent K-theory for all finite dimensional manifolds then you need an infinite dimensional Grassmannian.
However, the situation changes once you allow other model spaces (the largest category of such is the category of convenient vector spaces, the introduction of which is a very interesting read on calculus in infinite dimensions). There you can and do get much more interesting behaviour and you discover that you can study infinite dimensional manifolds as objects in their own right.
One example of this is the notion of semi-infinite structure. This is, almost by definition, only available in infinite dimensions (there are shadows in finite) and has proved an important source of ideas, if not actual techniques.
Of course, many of the techniques involve bringing things back down to finite dimensions in the final analysis but that's because we want to actually compute something and so end up with a number; and the easiest way to get a number is to count a finite number of things. But that's no different to any other computation, so shouldn't be seen as a disadvantage.
So back to the original question. Well, I don't really have a good answer to that because I work with infinite dimensional manifolds so I don't spend any time worrying about what others want to apply this work to, I just get on with it. But nonetheless, one theme that I see a lot is that of the infinite dimensional "picture" being the right one and the one that gives the intuition for how the finite dimensional approximations fit together.
Thus we see that Floer theory is really Morse theory applied to loop spaces, but not many will compute it as such. We see that the elliptic genus is (was originally!) really index theory applied to loop spaces, but again that's not useful for computations!
In general, anywhere where you've got functions that can vary, you've got an infinite dimensional manifold and it behoves you to remember it because it can give you important insights on how to proceed.