This article establishes a sufficient condition for Kobayashi hyperbolicity of unbounded domains in terms of curvature. Specifically, when $\Omega\subset{\mathbb C}^{n}$ corresponds to a sub-level set of a smooth, real-valued function $\Psi$, such that the form $\omega = {\bf i}\partial\bar{\partial}\Psi$ is Kähler and has bounded curvature outside a bounded subset, then this domain admits a hermitian metric of strictly negative holomorphic sectional curvature.