I'm normally loath to just quote Wikipedia, but it does have some relevant things to say:
The article on separabel spaces tells us that
every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset.
(cf densely defined operator) and that
many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces. Such constructive proofs can be turned into algorithms for use in numerical analysis [...]. A famous example of a theorem of this sort is the Hahn–Banach theorem.
Furthermore, the article on Hilbert spaces contains the following:
A Hilbert space is separable if and only if it admits a countable orthonormal basis.
In case of field theory, it states:
Even in quantum field theory, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman axioms. However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number of degrees of freedom and any infinite Hilbert tensor product (of spaces of dimension greater than one) is non-separable. For instance, a bosonic field can be naturally thought of as an element of a tensor product whose factors represent harmonic oscillators at each point of space. From this perspective, the natural state space of a boson might seem to be a non-separable space. However, it is only a small separable subspace of the full tensor product that can contain physically meaningful fields (on which the observables can be defined).