In his introduction to the trade paperback version of Batman Versus Predator, Dennis O'Neil said, "The only test I've ever found for judging the artistic credibility of using a character from one fictional universe in another such universe is this: If the characer were presented fresh, as a new creation, would he be acceptable?"
Using this standard, the only series where a crossover with either Alien or Predator would really work is Enterprise. With TOS and the series that followed it, the tech available to the Starfleet crew makes it far too easy to locate and neutralize the intruders.
I think this partly depends on your plot and the situation in which you place your characters. A landing party can indeed be placed in peril in any itineration of Trek just by virtue of the ship being elsewhere. The Galileo Seven is one example. Perhaps kidnapped crew might have to engage in a 'Running Man' style game featuring aliens or Predators. IDW placed a landing party on a world populated by zombies and I thought the story was quite fun - or at least fun enough that a two issue story wasn't enough for my tastes! There are plenty of other possibilities although TNG replicators would make a mockery of any 'limited resources' plot - that would be the first thing I'd destroy with acid personally!
Alternatively, a cloaking device can overcome Trek scanning or transporting technology or in my story I went for a 'natural' ability to avoid detection by scanner, which leads the crew to consider how they do this, which leads them to genetic engineering, which leads them to ask who was responsible for that genetic engineering, which leads them to the next plot point and so on.
The bar to these kinds of stories has more to do with the conscious decision to make modern Trek very family friendly rather than anything inherent in the nature of the franchise itself. This is why we mourn the loss of the parasites from TNG season one.
Squirrel Girl would totally dig Chakotay.