In this article, we study the relations of horizontally conformal maps and harmonic morphisms with the stability of minimal fibers. Let ${varphi}:(M^n,g){ ightarrow}(N^m,h)$ be a horizontally conformal submersion. There is a tensor T measuring minimality or totally geodesics of fibers of ${varphi}$. We prove that if T is parallel and the horizontal distribution is integrable, then any minimal fiber of ${varphi}$ is volume-stable. As a corollary, we obtain that any fiber of a submersive harmonic morphism whose fibers are totally geodesics and the horizontal distribution is integrable is volume-stable. As a consequence, we obtain if ${varphi}:(M^n,g){ ightarrow}(N^2,h)$ is a submersive harmonic morphism of minimal fibers from a compact Riemannian manifold M into a surface N, T is parallel and the horizontal distribution is integrable, then ${varphi}$ is energy-stable.