No. Let $X$ be the blow-up of $\mathbb P^2$ at two points, and let $f : X \to Y$ be the map down to $\mathbb P^2$. Let $D$ be the strict transform on $X$ of the line between the two points you blew up. This is a $(-1)$-curve, hence $h^0(X,D) = 1$. But $D_Y$ is a line in $\mathbb P^2$, which has larger $h^0$.
I might add $h^0(X,D)=1$ does not imply $D$ is extremal; you need to assume that $h^0(X,nD) = 1$ for all $n$. But this is a somewhat orthogonal question; it's not the issue in the example.