Suppose that $X$ is a compact Hausdorff space and take a sequence $(f_n)$ in $C(X)$ such that the ideal generated by $(f_n)$ is proper. Must this ideal be separable as a Banach space?
It looks to me that it should be the case. For instance this is the case where $f_n$ are indicators of isolated points in $X=\beta \mathbb{N}$.
EDIT: The answer is no if $X$ contains a non-separable open subset.
What if every subspace of $X$ is separable?
The answer is no. Let $D$ be the double arrow space and consider $D_1\sqcup D_2$, the disjoint union of two copies of $D$. This space is hereditraily separable. Yet the ideal generated by the indicator function of $D_1$ is non-separable.