Building on the premise that Equus is closer to a utopia than Earth, even if we assume mental issues don't affect creatures there on a severely incurable level, I think they would be able to recognize the symptoms if they saw them in person. They wouldn't know the exact names for said conditions, but they would be able to see the signs. Illness can be masked, but never completely hidden, at least in my experience. It affects things like body language, attitude, energy levels, behavior, and sometimes other physical signs. They would know something was wrong, even if they weren't sure what. The real question is how they would respond when they see how deeply these illnesses affect humans. They can only do so much to help, but it would probably be better than nothing. I'm curious to know everyone else's thoughts on the issue. This is just personal conjecture based on what factors I know.

7949962
Well, utopian doesn't seem like an appropriate word to me, idealized yes. It is a world of magic, many things are allowed, so one would think that it is possible to create an eternal kingdom of immortal creatures with magic. Well yes, it is possible to create such a thing, however from its conception it is condemned, when creating it is understood that it exists because it did not exist before. Therefore, a magical universe that allows the creation of immortality also allows the destruction of immortality. In the end, magic is synonymous with freedom. But not immutability. Everything changes, everything can have a counterpart.

7949962
I fear that you are just ignoring canon to make your ideal perfect society. Ponies have expressed severe symptoms of mental illness at several points of time in the show, but this is never addressed; I think it's safe to say that ponies are completely oblivious to mental illness.

7950460
It's been addressed, but there's a difference in how it's treated. They always go for the source because of how extreme it is. They sealed Cozy away because she couldn't be treated. Part of the reason for that is because no one knows who she is. She has no backstory to speak of. You can't treat what you don't know. There's always the possibility she could be treated later on down the line.

7949962
Well, just look at the show. There was the barking pony (Screw Loose, but I think that's just a fan name) in Read it and Weep, who was clearly a mental case, although she wasn't in a mental institution, just a normal hospital. If we go off of that, perhaps there just aren't enough cases of that in Equestria to have asylums and the like separately.
Later on, in Season 3, this pony has a cameo, and she seems much better off. Based on that, whatever the ponies do with mental patients work, and that's perhaps why you just can't see psychos walking around. And of course, because, MLP is a children's show, but let's ignore the meta for the sake of the argument.
Oh you were asking WHAT they do? Eh, don't have an answer to that