I'm not sure who to cast as the characters for this, but just for kicks and giggles, let's say Christian Bale plays Batman and somebody whoever can play Jason because it doesn't really matter.
This is a generalized summary of what I think a crossover between Friday the 13th and Batman could be like. It could be a comic mini-series, and an animated movie or Feature Film.
WARNING: TOTAL CHEESYNESS
Friday the 13th: The Darkest Knight
Jason has been defeated for sure in “Jason Goes to Hell” in 1993. Jason Voorhees is in Hell, unable to revive on his own accords. He needs outside influence to be revived.
In the DC Universe, The Joker is discussing another evil plot with Harley Quinn, in his lair. He reveals that through unnamed sources, he has managed to meet a Shaman that says he has ability to procure the ultimate enemy for Batman: a supernatural killing machine from another dimension. The only requirements are a living sacrifice when the killer appears, that must be someone close to him due to the severity of the ritual, and an object for which the killing machine may be revived through (a Horcrux in other words). It can be any random object, but the killer is tied to that object indefinitely, meaning that should the object be destroyed, the killer will expire as well.
Joker agrees to provide those requirements, and supplies a single Joker card as the “portal” for the killing machine. The Shaman also tells Joker that if he does not provide a sacrifice the killer will kill him instead, and that this killer is not the actual killer, but rather it is a “manifestation” of the killer. This means that Jason's mind and body can appear in the DC Universe, but the very core of his being still truly resides in Hell.
The Shaman performs the ritual for Joker, after providing him with a brief history of Jason, assuming he will sacrifice Harley Quinn. After a theatrical show of spiritual chaos, the manifestation of Jason Voorhees appears in the flesh. However, he is in suspended animation until Joker's sacrifice. When The Shaman questions him about this, Joker shoots The Shaman with a pistol, instead of Harley Quinn. The Shaman tells Joker with his last breath that he will pay for his betrayal, and Jason will surely kill him instead.
When the Shaman dies, Jason is released, and stands before Joker. Joker, having been prepared for this, throws Jason a comical flier, advertising Batman. He tells Jason,
“This man is your enemy, Jason my boy. His name is Batman! And he hates your mommy with a passion, and he frequently swims at Crystal Lake, in the nude!” This infuriates Jason and he attempts at Joker's life. Joker and Harley make a quick escape out the window of the building, parachuting to safety, and leaving Jason to do what he wants.
Confused by the brand new scenery of Gotham, Jason stumbles out into the night, machete in hand and masked with his trademark Hockey Mask. He tears apart a small gang of thugs as he steps out into the dark streets, and takes off into the night, searching for nothing.
Bruce Wayne is alerted by Commissioner Gordon that some awfully grizzly murders have taken place in the slums of Gotham lately. The murders took place in a straight pattern, from Jason's origin to a certain abandoned part of town; seemingly, the killer (Jason) stopped and is now hiding out.
Batman takes off to find Jason in the abandoned part of town. After some hard-pressed searching and detective work (a major section of the story), he hears from a thug that Jason is now in the basement of an abandoned building nearby. Batman goes to this building, and sure enough stumbles upon none other than Jason. But Jason is trapped, he fell through a weak floor, and now the water pipes have partially flooded the room. Jason is trapped in a corner.
Batman uses his batarang and attaches it to Jason. He zips Jason and himself to the top floor, but in no time at all Jason was already trying to kill him. Cutting the zip line, Jason gets to his feet and launches at Batman with fury. Batman manages to dodge his machete, and land some pretty hefty kicks in the process, but nothing has fazed Jason. After a chase scene, Batman is no longer in reach of water. Realizing the issue at hand, Batman leads Jason into a warehouse, to attempt at Jason's defeat again. Batman and Jason fight, and Batman is repeatedly injured until Jason finally cleaves Batman in the side. Now mortally wounded, Batman fears he will not make it out alive if he does not kill Jason soon.
When all seems lost, Joker shows up in the nick of time. He reveals to Batman that he has been following him secretly, and now he intends to get rid of Jason, because he wishes for Batman to die by his hands and not Jason's. Angered by Joker's appearance, Jason goes after Joker. Joker reveals a missile launcher, and shoots a missile at Jason. He nearly misses, but the blast sends Jason flying, and Jason appears dead when he lands (we are to assume he is not dead, seeing as the only way he died was when he was completely blown apart in "Jason Goes to Hell"). Unable to attack Joker, Batman pleads that Joker reconsider his intentions to kill him because it would be dishonorable to kill Batman in this state of weakness.
During their discussion, Jason has come back up. Without revealing the fact to Joker that Jason is nearing him, Batman intends to let Jason kill Joker from behind. Seconds before Jason gets his hands on Joker, Harley Quinn shows up because she was following Joker as well. To save his life, she burns the Joker playing card with a lighter.
While the card falls to the ground in ashes, demonic hands erupt in a fiery light of Hell to bring Jason back. Joker and Harley are laughing hysterically, but the satisfaction quickly disappears when another portal of demonic hands and fire pop up to pull Joker down too. The Shaman appears out of nowhere, and tells him that because of his betrayal, Joker inadvertently made himself the sacrifice, because the summoner must be honorable enough to suffer the sting of defeat. Because Joker proved himself dishonorable and unsuitable to summon Jason Voorhees, he and Jason both are dragged to Hell.
As Joker's hand sinks into the ground, he laughs heartily and the echo of his laughter is heard as he is finally dragged into Hell completely. Harley Quinn is distressed and heartbroken, and Batman leaves her as she begs him to take her in. When Batman leaves, The Shaman appears again to Harley. He reveals that he is Freddy Krueger, a dream invader from another dimension, and he promises to Harley that he will avenge Joker's death if she lets him kill her. She agrees, and Freddy takes her life.
Having successfully entered the DC Universe, Freddy has defeated the only other people crazier than him: Joker and Harley. This could open up to a sequel where Freddy torments Batman!
It's rough, I know. But a gritty and scary animated movie like this would be astronomically phat.