Gotham City has been plagued by some of the most dangerous and insane criminals in all comic history, but even Batman knows that the Penguin doesn't belong in Arkham Asylum. A long time ago Batman admitted that of all his enemies, Penguin belongs in actual prison, not a mental asylum.
Batman is no stranger to battling villains who suffer from psychological problems, from characters like Jervis Tetch, who believes himself to be the Mad Hatter from Wonderland, or Victor Zasz, a horrific nihilist who believes by killing people he is freeing them from the burden of life. This isn't even to mention characters like the Joker or Two-Face, who are obviously suffering from their own mental issues. Batman is so used to dealing with insane people that he once assumed Catwoman was a kleptomaniac due to her penchant for stealing. But despite all the lunatics Batman has faced, he does recognize that one iconic villain isn't insane at all.
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Batman's true feelings about the Penguin are made clear in 2013's Detective Comics #18 by John Layman, Jason Fabok, and Jeromy Cox. While preventing a riot in Arkham Asylum, Batman runs into the Penguin, who was brought there by the Joker due to the Batman: Death of the Family event that was going on at the time. After Batman has subdued all other criminals, the Penguin protests that he hasn't done anything and he doesn't belong in Arkham. Surprisingly, Batman agrees with Cobblepot, thinking that Penguin instead belongs in Blackgate Prison. Penguin once beat the Bat-Family single-handedly and has been as much a thorn in the Dark Knight's side as any other foe. But the big difference between Penguin and the rest of the villains Batman often fights, is that Penguin isn't actually insane.
Most of Batman's rogues' gallery are compelled to do what they do due to some form of comic book interpretation of mental illness. Two-Face is said to be suffering from an extreme version of schizophrenia and Dissociative Identity Disorder. Riddler has a very serious case of Obsessive Compulsive Disorder and an inferiority complex. But Penguin? Penguin by all accounts is perfectly sane. He is slightly physically deformed in some interpretations, but there's nothing mentally wrong with him. Penguin is a criminal and a gang lord, sure, but he's entirely in a sound state of mind. He isn't compelled to leave riddles to prove how smart he is, he doesn't have a separate personality urging him to do bad things. Penguin may have one bad day here and there, but he isn't a lunatic. Compared to the other villains in Batman's rogues' gallery, Penguin is entirely reasonable.
Ultimately, while Penguin is a horrific person who has killed and stolen to get where the riches and power he has today, that doesn't put him on the same level of Batman's other villains. Certain characters like Two-Face and Riddler can't really help the evil they do in their current state of mind, which is why they belong in Arkham Asylum, to get the help they need. Penguin on the other hand, is completely aware of his actions and how they will affect the lives of other people. This is why he doesn't belong in Arkham Asylum, out of all of Batman's villains, Penguin belongs in prison.