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.
[Edit:] here is a proof that if a noetherian algebraic stack $\mathcal X$ with finite inertia has the property you want it is tame with affine moduli space. Let $\mathcal X \to M$ be the moduli space. Let $\mathcal G$ be the residual gerbe over a closed point of $M$; then $\mathcal G$ is closed in $\mathcal X$, so the cohomology of each quasi-coherent sheaf on $\mathcal G$ is trivial. The moduli space of $\mathcal G$ is the spectrum of a field, so $\mathcal G$ is tame. This implies that the automorphism group of an object of $\mathcal G$ is linearly reductive. One of the results in the paper implies that an open neighborhood of $\mathcal G$ in $\mathcal X$ is tame. Since every non-empty closed subset of $M$ contains a closed point of $M$, this implies that these open neighborhoods cover $\mathcal X$, so $\mathcal X$ is tame.