There really is no incentive in New Eden outside of RP/epeen reasons to fight player pirates.
This is 100% true.
But so is this; There is no incentive in New Eden outside of entertainment value to
do anything at all.
I'm not really sure it can be said that piracy is supported by any actual mechanics - after all the only pirate-like activity - Ransom - is entirely created by players. The rest is fully available anywhere else too; boat-violenceing. Piracy has several consequences - sentry aggro for agression, sec-loss, eventual lock-out from high-sec (Avoidable in pods for the sake of suicide-assaults or whatever, but that's another topic) and flashy you-are-now-free-game-and-the-navies-hate-your-ass status anywhere. Much like the trade/transaction mechanic that allows money/item trading being used for ransoms however, players have found ways around this to support their desired play-style.
While I am not sure piracy has a direct mechanic to support it, I do know that anti-piracy definitively don't. What is the AP's job? In short - oppose piracy. The only direct means for that is inconveniencing them and violencing their boats in place of CONCORD - in High-sec you get poped for agression, in low-sec you lose sec-rating until other players can do CONCORD's job for them. ergo, the term 'vigilante' or more common - anti-pirate.
There is no mechanic that positively flag you as a pirate - if you join a corp as the 10th toon and 6 of the others are perma-flashers you don't get punished for association by instantly dropping to perma-flash yourself. This enables you to scout anti-pirates for your friends - AP's won't shoot non-flashers, if they do shoot you, they will have to spend time fixing sec-rating regularly. Time your friends can use performing more piracy while the AP's are in high-sec grinding standings.
There is no NPC entity that flag you as a 'pirate' (The term is entirely player made, like it's counterpart) - there is no CONCORD 'militia' vs the various pirate 'militias'. Ergo AP's have to use their own discretion and decisions have to be made on the run as to who to shoot or not. Alternatively if there are many non-flashies in the corp/alliance with all the pirates, a war-deck allows them to use lighter ships and drones near gates and stations, and none of the non-flashies get any sec-hits for helping them. Ergo, a double-edged sword far as the outlaw mechanic is concerned.
When a pirate and an anti-pirate gang is out roaming, the pirates can fire on whatever they want - what do they care for sec-rating they don't have? They can do this whenever and however they want, and decide not to if, say, the AP gang they ran into is to powerful. Should the AP's fire first, IIRC the pirates are free to return fire because the AP's gain temporary flashy status as well. In fact, should they fire on the pirates anyone can fire on them in return with impunity, and any sentry in range will not fire on either the pirate who got shot, his attacker, or anyone who now attack his attacker. (Correct me if I recall this wrong.)
AP's will have to contend with the fact that any known pirate associate (corp or alliance member, known helper etc.) can chose to make a run for it if they don't like the odds, and the AP's will take sec-hits for agression and possibly ship destruction as a consequence of attacking this person, so long as he has not dropped below -4.9 and/or agressed recently. Gaining a bit of sec-rating back can easily be done with low-sec ratting or missioning to stay above the -5.0 line and force AP's to chose between the kill and their sec-rating. This condition can also be exploited by swiftly making a scouting toon in a noob ship to keep eyes on any AP's, letting you know who, how many, in what, where, doing what, at all times. Ofc this can also be done in high-sec during wars, but in low-sec the AP's can either let the pirates gain risk-free intel or lose sec-rating for poping and podding a disposable scout. Any attempt to do this for AP's requires the 'disposable' alt scout to atl be cloaked because otherwise the pirates will just kill and pod it.
I'm sleepy, so I'm sure I've forgotten some factors to the pirates vs anti-pirate 'conditions' of EVE and the mechanics behind the whole thing.
So, with the mechanics lacking as they are, anti-pirates (and everyone else who live in high-sec) have several reasons not to care with attacking pirates and only one incentive to do so; entertainment. Entertainment marred by an opposing force that has the upper hand in nearly all situations. Still, some persist, and don't end up just going pirate while they are fighting them.
Let's assume a corp desires to fight piracy as a primary objective, just because. They wish to live in high-sec and have access to it, let's say for the sake of deccing known pirate-helper corps, or RP, or whatever. How can they go about consistently fighting pirates, overcoming the limitations of the game, and still maintain a specific presence in low-sec that gets a satisfactory number of pirate camps busted, ships killed and systems cleared?