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 also 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.

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.