Usually the ring of adeles is defined for number fields: if $K$ is a number field the ring of adeles of $K$ is:
$$\mathbb A_K:=\prod_{v}' K_v \;\;\;\;\;\;\;\;\;\;\;\;\;(\ast)$$
where $v$ ranges among all valuations (archimedian and non-archimedian) and $\prod'$ is the restricted product with respect to the valuation rings $\mathcal O_v$. With $K_v$ I mean the completion of $K$ with respect to $v$.
Now suppose that $K$ is an algebraic function field over $k$, is it true that the adelic ring is defined as in the equation $(\ast)$ but with the difference that $v$ ranges among all valuations which are trivial on $k$?
I think that we need to add this restriction on $v$ in order to preserve the correspondence with the closed point of the projective curve $X$ such that $K(X)=K$.
Assuming that $k$ is algebraically closed in $K$, a valuation of $K$ is indeed trivial on $k^\times$. Please see page 13 of here for a detailed description.
However, usually when constructing the adeles, $K$ is assumed to be a global field. A global function field is defined over a finite field $k$, in which case the triviality of the valuation on $k^\times$ is automatic.