Your question is the famous "separable quotient problem", as Ady mentioned. From here on, "space" means "infinite dimensional Banach space". A space $X$ has a separable quotient provided
$X^*$ has a reflexive subspace (obvious), a subspace isomorphic to $c_0$ (Rosenthal and me), or $\ell_1$ (Hagler and me). A result of PANDELIS DODOS, JORDI LOPEZ-ABADAND LOPEZ-ABAD and STEVO TODORCEVIC is that it is consistent with ZFC that if $X$ has density character at least
$\aleph_\omega$ then $X$ has a separable quotient; see
Every dual space has a separable quotient (Argyros, Dodos, Kanellopoulos):
There are other striking things that I can't locate quickly.
Every non reflexive quotient of a $C(K)$ space contains a subspace isomorphic to $c_0$ (classical result of Pelczynski), so $\ell_1$ is not a quotient of $\ell_\infty$.