No, I do not care if this is cliche I'm doing it anyway. First off the bat is one James T Kirk.
Jim is an earthbender. For starters, a lot of Earthbending is based on indomitable will, with sheer dedication powering their attacks. Not many people have more willpower than Jim Kirk. But to add to this, Earthbenders can run up a reputation for being more brawn than brain, relying on brute force all the time, and Jim has a similar reputation for using sheer force.
Despite these reputations, the best earthbenders (see: Toph) are actually very strategic, waiting and watching for the perfect time and way to strike. Similarly, Jim is actually very good at strategy, and can use his opponents presuppositions about him against them.
Furthermore, if Jim was an earthbender, then he could also be a metal-bender. This would allow him to be even closer to the Enterprise on a near-spiritual level, able to really connect with her intimately. Since Jim is so obsessed with the Enterprise, this seems to be appropriate.
Next up is Bones. I honestly considered firebending here, but I have to go with waterbending. After all, that's the one with healing powers! Besides just that, though, waterbenders are very flexible, and do a great job of adapting to situations. They also tend to avoid brute force in favor of redirecting attacks and using more circuitous methods of fighting. Further, waterbending might have the most variety in what it can be used for, meaning that creativity is paramount of effective bending.
Now, we know McCoy is very adaptable, even if he hates the situations he finds himself in (see: all of space) he does a great job making the most of it, curing aliens he's never seen before and creating cures on the fly. He isn't into brute force, preferring to be sneakier and get in a good hypospray instead of throwing hands. And finally, as a scientist he would be very good at finding new, unique ways to use his waterbending skills.
This brings us to Spock. I went back on forth between water, earth and firebending for him, but I finally settled on firebending. First of all, it allows him to be McCoy's opposite, which is nice. But more than that I think it presents great world-bending and character-exploration opportunities.
I thought about earthbending because Spock is also very strongwilled, and I thought about waterbending because it is a very graceful element, and it seems to lend itself well to Spock's scientific way of thinking. However, I think that firebending does a better job of getting to the core of his character.
After all, Vulcan's are not naturally controlled, they are extremely emotional and passionate, and firebending would reflect this well. In fact, I think it would be cool if all Vulcans were firebenders, or at least if that was the only kind of bending they had. Not only could all Vulcans firebend, but they would naturally be extremely powerful at it, naturally summoning white fire. However, in the before-Surak times, they almost killed themselves with firebending and emotions, and now both of them are banned.
Technically they can still bend, but firebidding is forbidden, and you can be cast out of Vulcan society for actually using it. This means that Kirk and McCoy may not even know Spock can bend when they first meet him, because Vulcan firebending isn't exactly advertised anymore. Eventually, however, Spock will have to make the choice between being a good Vulcan, and firebending to save his friends… and we know what he will choose.
Firebending could actually be very dangerous to Vulcans, because you have to be emotional to do it properly, which can lock them into a viscous circle of emotion with the emotions and fire feeding each other.
As for the not-the-big three:
Scotty is technically an earthbender, but he can't actually bend earth, only metal. A true engineer!
Chekov is a firebender - he's just a firecracker always getting in over his head.
Sulu is a waterbender - he likes the elegant art of redirection, but he's also crazy enough to get into the esoteric forms.
Uhura is an airbender - she doesn't let anything keep her down, and she works to avoid direct confrontation but when she needs to holder her ground she can come at you from multiple angles at once.
Finally, I had a commenter ask me what I think about Klingons. I gotta say that I think they should be earthbenders. I considered firebending, but I think firebending is a little less brute force than the Klingons generally get credit for.
If you look at the Klingon combat forms and ritual dances, especially in TNG, the forms are very similar to earthbending. Klingons are also known for being extremely stubborn, which would fit very well.
Despite this, I don't think Klingons would be able to metal bend, because that focuses on small impurities, and I think Klingons would instead be good at large scale attacks - large amounts of rock, large amounts of force. I think of Klingons as more of a broadsword than a rapier, and I don't think they would be very good at manipulating small amounts of earth. Very skilled Klingons could manipulate small amounts of earth, but even they couldn't metal bend. That said, an unusually high proportion of Klingons should be able to lavabend compared to other earthbenders. After all, it seems like a good fit for their firey personalities!