First of all, most cross-overs are primarily designed to be one-shots, rather than an ongoing series. Thats probably why they're fascinating or interesting, to at least some people. The trick is to regard them as Alternate timelines. All starts with the famous "What if?"
Theres been a few crossover comics for Terminator:
Terminator / Robocop - that one actually worked alright, see'ing two totally different types of metals going head to head.
Terminator / Superman & Justice League - Interesting for a one shot, but thats about all.
Terminator / AVP - That one got a bit weird, and it featured Cal from A4.
What about Terminator / X-files? (ok, ep 2x13 had a nice Xfiles feel to it) perhaps write one from the Mulder & Scully POV. Mulder getting mixed up in a gunfight & chase with a Terminator, and is convinced its some alien hybrid experiment rather than a robotic endo-skeleton? Scully could also investigate with Mulder the UFO convention, and take interest in Sarah's observations. It would be kinda re-writing Ep 2x13 as an Xfiles episode, leaving out Cam, Derek and John.
Have you tried Terminator / GI Joe as an idea? The Joes get called in to find and protect somebody (government official), from a relentless stalker & killer. Except the killer is not even remotely something of Cobra's typical creations. Through the grapevine, Cobra Commander gets informed about this mechanical menace and seeks it out to strike a deal with it, offering to build the facilities that'll become Skynet. Of course, Cobra Commander has his own plans, to use Skynet and its future technology for his own personal world domination. As a preventative measure, Skynet sends back a few squads of Terminators (who are far superior to Cobra's own B.A.Ts - Battle Android Troopers) to oversee and make sure Cobra Commander honors his agreement. As things cool off for the Joes, once they deal with the Terminator robot, they take it back to the PIT for closer examination. They extract from its memorybanks details about Skynet and the future. And what a surprise they get when they head over to Cobra Island...
Or... You could try doing a post J-Day style GIJoe type story. The GIJoes legacy still alive and kicking in the the future, many of which sons and daughters of the original Joes, still operating from within the PIT, trying to hold their own against Skynet's forces. Cobra dosn't even have to factor into this one, as a power-hungry organization, it would've collapsed from power struggles. The ol' Cobra Island remains could still prove useful however.
One of the other things about cross-overs IMHO, is choosing which genre to serve as the 'host / backdrop' for the dominant setting, and which genre to bring in as a guest to mix in.
With the first GIJoe crossover example, we'd be bringing in the Terminators in as the guest, as well as the future J-Day event yet to happen in the Joe's world. In the second Joe example, we jump right into the Terminator's J-Day post apoc world, with the Joes a new generation rag-tag band spurned from the originals, and any of the original suriving Joes old men and women now. The PIT would be more like a survivor's shelter than a pristine conditioned underground garage and HQ that it once was. All the equipment old, some nolonger functional, supply of spare parts comes salvaged from wrecked vehicles. Once SkyNet locates and attacks the PIT, the group (those who survived) would have to flee and find new shelter. Cobra Island comes to mind. There'd be alot of similar skirmishes as we seen in the T movies, Joes on cobled together vehicles fighting tank and aerial HK's. Joe vehicles like the AWE Strikers, Desert Fox, and few others come to mind.
I used to play in a SW / Robotech / Macross crossover rpg game, and for similar reasons, we picked the SW universe as the host to play in , and brought in the Megaroads fleet in as guests through a mis-fold which ended up in the SW galaxy.
A Stargate / Atlantis / Terminator crossover could be interesting. Scratch the Gould' (SP?) for this particular occasion and the SG-1 / SG-A team travels to a earthlike world not much different than their own. Or maybe they did return home, but some anomally intersecting their return trip propelled them to post J-Day, they arrived back in their mountainside bunker, find its deserted and dusty. Some uniformed skeletal bodies strewn about, and stepping outside proves to be desolate and barren. Perhaps they run into a small resistance vehicular attack group thats been fleeing and licking their wounds from a supply raid gone Terminator ambush. Lots of oppertunities to pickup and go from there
Terminator / Firely - Mal's crew get shot down and end up having to land on a Confederate planet. Except something's gone horribly wrong. The cities are in ruins, and theres metal machines all around. Mal's crew has to find parts and fuel to fix their ship before they attract Skynet's attention. Jayne and Co could have a few hair-raising run ins with the HK's and come across some resistance fighters. (I could just picture Jayne going to town unloading a heavy clip of plasma into some aerial HK's while riding on the back of one of the resistance's junker trucks)
Terminator / Resident Evil - Skynet's already incorporated into the US Military DOD (Department of Defense) network. Someheres deep underground in Ukraine, Umbrella Corporation builds a new laboratory. An accident happens and the T-Virus escapes, rapidly infecting the surrounding towns. Through the media, Skynet learns of the outbreakd and immediately launches its ground nukes at Ukraine, predicting the nuclear blast radius should elimimate the threat. However, Moscow detects the launch as an unprovoked attack and launches its own nuclear arsenal. Moscow's strike hits the US, and the attack ruptures one of Umbrella Corp's underground labs causing a T-virus outbreak that rapidly fans out through the states. Skip ahead 20 some years - the battleground in L.A is a 3 way fight. The surviving Resistance, the Terminators, and the infected zombies of all forms - humans, rodents, etc.
Terminator / Resident Evil - Alternative Variant - Pretty much the original premise of the J-Day happens as it did in T1 and T2. Cut to 20+ yrs later, with the Resistance fighting Skynet's Terminators and trying to survive as humanity on the endangered species list. The twist happens when a few Resistance scouts stumble upon an access to an underground bunker / Lab which they initially think may be the heart of Skynet. Its not. They find the facility deserted and somewhat smashed up. They leave to bring a few more fellows to checkout the place, figuring it'd be a perfect new base of operations. But one of the members gets infected unknowingly from the powder like dust. The rest of the team follow suit with the infection and succumb to a zombie like state. They're not heard from in over a week, a party is sent to investigate and once unleashed, the virus starts spreading like a slow hatching epidemic. Some infected Resistance members get captured by Skynet, and Skynet's own camps of human workforce get infected. It becomes all out chaos. Even the vat engineered Terminator skin gets infected.