Let $X,Y$ be Hausdorff spaces and consider the compact-open topology on $C(X,Y)$. Let $F\subseteq C(X,Y)$ be a non-empty subset and consider the topology $\tau$ on $C(X,Y)$ with subbase $$ B_F\triangleq \{f\in F: f(K)\subseteq V\}_{K \mbox{ compact in X}, U \mbox{ open in Y}}. $$
Is $F$ dense in $C(X,Y)$ with this topology?
Yes. Much more generally, let $Z$ be any set, let $F\subseteq Z$ be nonempty, let $B$ be a set of subsets of $F$, and let $\tau$ be the topology on $Z$ generated by $B$. Then every element of $\tau$ is a union of finite intersections of elements of $B$. All of the terms of such a union intersect $F$ unless they are empty, so any non-empty element of $\tau$ intersects $F$. That is, $F$ is dense with respect to $\tau$.