Since nobody gave a reference yet, in my paper "Artithmetic moduli of generalized elliptic curves" I included a proof that an Artin stack whose geometric points have trivial automorphism schemes is necessarily an algebraic space. See Theorem 2.2.5(1) there; I am sure this is a folklore fact (which I inserted there because I didn't know a reference, and to my surprise seems to not be stated in the L-MB book). So that answers the original question: if the moduli problem is an Artin stack and a coarse moduli space exists then it is a fine moduli space (meaning that the moduli problem is an algebraic space) if and only if objects over algebraically closed fields have trivial automorphism schemes (stronger than just trivial automorphism groups!)groups, except in the DM case when equivalent since then such groups are etale).
I wrote that paper in the days before I realized that non-qs algebraic spaces made sense, so I had the convention throughout (following the L-MB book) that diagonals are separated and especially quasi-compact. I have not revisited the proof to see the effect of weakening these assumptions (especially the q-c assumption) on the diagonal. I should do that some day.