Can you sacrifice a creature to pay a cost after that creature has been targeted by a destroy spell?

When a spell like Murder is cast targeting your creature, it goes on the stack. At this point, before it resolves, both players receive priority. If you have a way to sacrifice the targeted creature as a cost (for example, to activate an ability), you can do so in response, before the destroy spell ever resolves.

Costs are paid immediately when you activate an ability or cast a spell (CR 601.2h, 602.2). Sacrificing a creature as part of a cost removes it from the battlefield instantly — it does not use the stack. This means the creature is gone before the destroy spell resolves.

When the destroy spell later tries to resolve, it will look for its target and find it is no longer on the battlefield. Per CR 608.2b, if all targets of a spell are illegal at resolution, the spell is countered and does nothing. The destroy effect never happens.

Concrete example: Your opponent casts Murder targeting your Doomed Dissenter. You respond by activating Ashnod's Altar, sacrificing Doomed Dissenter to add 2 colorless mana. The Doomed Dissenter dies as a cost, its death trigger goes on the stack, and when Murder tries to resolve, its target is gone — it is countered by game rules.

Note: if the sacrifice ability itself has a cost that requires targeting (not relevant here), or if the destroy spell has multiple targets, the rules may differ slightly. But in the standard scenario described, sacrificing in response is completely legal and effective.