I see the following statement in the proof of the local structure theorem of DM stacks from Jarod Alper's notes (Theorem 4.2.1).
My question is why an etale from an affine scheme to a separated scheme is affine. Furthermore, is such an etale morphism still affine if $U$ is not affine or $\mathfrak{X}$ is only quasi-separated?
I tried to prove the statement by using the cohomological criterion for affineness or by assuming the morphism $U\to\mathfrak{X}$ is standard etale. But neither seems to give a proof in an obvious way.
I do not have enough reputation to write a comment so I will write it as an answer. Any morphism from an affine scheme to a stack with affine diagonal is affine. I believe you can find a proof on the stacks project. A separated DM stack has finite diagonal.