The degree map gives a short exact sequence $$ 0 \to J(C) \to Pic(C) \to \mathbb{Z} \to 0,$$ where $J(C)$ denotes the Jacobian of $C$. This is an abelian variety of dimension $g$, where $g$ is the genus of $C$. In particular, $Pic(C)$ is always finitely generated when $C \cong \mathbb{P}^1$.
To deal with the case of higher genus, you might as well ask: for which fields $k$ do abelian varieties over $k$ of positive dimension have finitely generated groups of rational points?
Clearly this holds for finite fields, and the Mordell-Weil theorem also says that this holds for number fields. More generally, this is known to hold for fields which are finitely generated over their prime field.