There are two ways to frame this.
1: "Why is the monster targeting a downed PC?"
2: "Why isn't the monster targeting a downed PC?"
Regardless of how you frame it, the answer always depends on the monster, and the DM's interpretation of that monster's behavior. But the tone is important.
Unintelligent monsters, like a plant or an ooze, might not know the difference--to them, a downed character is just an easier-to-eat pile of organic matter (easier than the other piles of organic matter that keep moving around and stabbing, anyway). Intelligent monsters know that downed characters don't stay down for very long, and will taking extra steps to ensure they don't get back up again. Cowardly monsters might see a downed character as an opportunity to escape. Exceedingly cruel monsters might wish to capture the downed opponents to take home and torture, prolong their suffering. Greedy monsters might take prisoners to hold for ransom (or to sell), sparing the downed opponents since they are less valuable dead. Alien creatures might spare the downed opponents to experiment on, or to incubate their eggs, etc.
But none of this is going to matter if the players can't trust their DM. Without that trust, the players will always feel that the DM is unfairly trying to kill their character out of spite or malice. So it's important that when you (the DM) decide to target downed PCs, you use plenty of story and "gloating villain monologue" to explain the situation. If the party realizes the BBEG is motivated by greed, they might be able to distract it with money long enough for the cleric to save their fighter.