I tend to try to work out what the monsters would do.
Most creatures will switch targets if someone goes down, because they feel threatened by them. There are two cases where I would have the monsters actually attack damaged enemies:
1) Anything motivated by hunger that doesn't know about or isn't threatened by the party. Because the players reacting to something trying to eat one of them is a nice motivation.
2) Intelligent, tactically trained or lead enemies who have seen the party using in-combat healing to bring someone back up. In that case, I'll happily have them make one extra attack to a downed PC. I telegraph this clearly the first few times a party sees it, though - one of them shouting something like "They've got a healer - if they go down, make sure they stay down". The party learns to expect it after a while.
The death save system is something I'm very glad was imported from 4e - unlike the 3e system of -10 and you're dead, it means I as the GM feel that it's perfectly reasonable for me to damage a downed PC if the situation calls for it.
I like both of these - the idea of a big, slavering monster taking someone down to zero HP, then licking it's lips and salivating because next turn it's going to eat them should give any player the willies. Should encourage much panicked healing.
Intelligent, nasty and possibly dishonourable NPCs should definitely do that. I also suspect that almost any opponent with Multi attack who can't get to another PC could well use their extra attack on the downed PC, just to be sure.
The 5E Death Saving throw system is great - it improves on the already good 4E system, with the stabilizing after 3 successes, and the standing up on a roll of a 20/two fails on a roll of a one. 4E didn't have either of those, and failed death saves didn't go away until you took a short rest. There were a few monsters in 4E that could force PCs to take Death Saves with their attacks - they always added a certain level of worry to any fight they showed up in. Not entirely sure how to replicate that effect in 5E.