Let $S$ be a local scheme with closed point $s$. Let $x$ be a point in the fiber of $s$ along an étale morphism $X\to S$. Under what conditions is the extension of residue fields $k(x)/k(s)$ algebraic?
Lemma 28.19.3 of the stacks project says that in the locally finite type case, $k(x)/k(s)$ is finite iff $x$ is closed in the fiber of $s$. This does not assume $s$ is a closed point.
First of all, since you are asking about residue fields in the fibre over the closed point $s$, you can just replace $X \to S$ by its base-change $X_s \to \operatorname{Spec} k(s)$ via the canonical closed immersion Spec $k(s) \to S$, and so your question becomes: if $k$ is a field and $X \to $ Spec $k$ is etale, for which points $x \in X$ is the residue field extension $k(x)/k$ algebraic? The answer is: for all of them, as Hoot's comment indicates. (They are also all separable. Indeed, a morphism $X \to $ Spec $k$ is etale precisely if the domain is a disjoint union of $k$-schemes of the form Spec $l$, where $l$ is a finite separable extension of $k$.)