My take is that the Punisher targets and kills those criminals who don't give damn what harm befalls anyone else as long as they get what they want./ That was certainly Ennis's take in Punisher MAX; Frank says as much to his naighbor int he flashback from the first arc, and again to the mobsters who (in MAX continuity) killed his family in LThe Cell.
His primary target has always been organized crime, thoigh. If you're selling pot and the money's going upstream to a violent cartel, and you *know this*, then Frank will likely kill you. If you're some doofy teenager selling it to his friends, he'll make you *think* he's going to kill you to find the next guy up the chain, but he probably won't *actually* kill you.
He will kill anyone other than a police officer who tries to kill him. He also won't kill a police officer under any circumstances, even a very corrupt one, for mostly tactical reasons. When he did so in Jason Aaron's MAX series, it was an indicator of how far Wilson Fisk had pushed him.
Aside from the time Jigsaw drugged him, Frank has pretty much never killed anyone who was not actively or willfully malign. That is, he tends to target criminals who know the damage they're doing and just don't care, or "mad dogs" who are causing mass pain, carnage, and suffering.
He doesn't do this because he imagines himself a noble protector of all who are innocent and good, of course, but rather because that's how he can keep going and justify his continuing war to himself.
In the late 80's Frank experimented with a little bit of a softer edge at the request of Micro, and used none lethal methods as well as letting certain criminals who he would probably normally execute escape - mainly due to issues such as their age if they were very young. The first story arc of 'War Journal' Vol 1 depicts this new attitude.
For the most part however Frank has been pretty unmerciful towards serious criminals regardless as to any other factors involved, although there are a few other examples where he has let criminals live, usually because he needed something from them, and he has been known to work alongside criminals such as Micky Fondozzi, and in the ill fated and short lived second Punisher ongoing series where he joined the mob in order to reform them from the inside.