As seen in some contributions here, non separability of a Banach space can lead to pathology in the properties of measures or measurable functions with values therein.  As an example of this phenomenon, I would mention Stegall’s result that a non-separable dual of a separable Banach space fails the Radon Nikodym property.  The construction that he used even charactises dual spaces $E’$ with RNP as those for which separable subspaces of $E$ always have separable duals.  The details are in the classic text by Diestel and Uhl.