There are differences between anti-heroes and villain protagonists. One of the better sentences about anti-heroes: "They practically do the right things for the wrong reasons and wrong things for the right reasons". While an anti-hero may do bad things, it's typically for a good reason.
The Punisher is a murderer, but typically focuses on criminals that are worse than he is and doesn't run around gunning down cops and civilians while declaring himself to be God. He's also often portrayed in either a sympathetic or positive light; sure he's a vigilante and a killer, but he stopped that child rapist from hurting more children, so yay! Even if you consider him a villain, there's no way you could argue him as the worst villain in his series.
EDIT: Rambling spoilers for Death Note and PS2 Grand Theft Auto games in Message Detail. I always forget we can do that.
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In this world which will someday fall to ruin...
...dreams and hopes don't matter at all.