Suppose that $\mathcal X$ is an algebraic stack with finite inertia (for example, a separated Deligne-Mumford stack); then, by a well-known result of Keel and Mori, there exist a moduli space $\pi \colon \mathcal X \to M$. The stack $\mathcal X$ is called *tame* when $\mathrm R^i\pi_* F = 0$ for every quasi-coherent sheaf $F$ on $\mathcal X$ and every $i > 0$. From the definition it follows easily that tame stacks with affine moduli spaces have the property you require. In characteristic 0, an algebraic stack with finite diagonal is tame if and only if it is Deligne-Mumford.

There are several different characterizations of tame stacks; see the paper "Tame stacks in positive characteristic" by Dan Abramovich, Martin Olsson and myself. Using the results in that paper, it is not hard to show that a noetherian algebraic stack with finite inertia has the property you want if and only if it is tame with affine moduli space.