First off, I want to get something off my heavily-armored, bullet grazed chest.
1. The Banshee is an aircraft. Keep it in the air. It is not a ghost, it is not a warthog, it is not meant to run people over. It is meant to fly.
2. Don't hold a grudge against the banshee just because you enter a game and the moment you spawn you're run over by a banshee noob. Get even. Don't hate banshees, hate the stupid banshee pilots (not worthy of the title of banshee pilot) that think they're the first person to have the idea of running people over with something that's supposed to stay up in the sky.
Now that I'm done with that, here are the real tips.
1. I cannot tell you how many banshee pilots I have gotten with the shotgun. When I use the shotgun against banshees, I realize why people use such inaccurate, unfocused fire arms against ducks when they go hunting. Because the pellets spread out in the shotgun reticule decently at medium range, I use it all the time when I sense banshees on the horizon, at medium range. But the shotgun really shines when the banshee pilot goes right after you, better yet, when the noob tries to run you over. There are weapons that work just as well (in some cases even better), such as the pistol, and, if you're a decent shot at leading your weapons, a rocket launcher.
2. As I mentioned before, unless you want to become the stinking, putrid laughingstock of the Halo gaming community, don't run your banshees into the ground unless you want to be taken out by a decent shot. Stay high in the air, and when you spot your target, plummet towards the target, pressing forward as you go. Whether your target is in a vehicle or on foot, shoot your fuel rod cannon. Flying to the ground will give your shot an extra boost in accuracy. If your target is weakened, he will probably die. If not, finish him off with plasma guns. Or plasma guns first, then fuel rod cannon. With moving targets, it will take some practice, but you will get better. (I suggest you stay away from the LAAGs of the warthogs, unless you're feeling lucky, blast the hog upside down with the fuel rod and finish your enemies off with the second shot or with plasma bursts.)
3. Never hover. Unless that target is about to score the flag or take the skull, and there's no one else around, don't take the risk. Any decent shot with a rocket launcher, a pistol, a tank, a warthog gun will eat you for breakfast and crap your noobish remains out when you respawn.
4. Banshee versus banshee battles are very tricky. I usually lose in these, but I can give you a bit of advice about this. You and the enemy have a ninety percent chance of chasing each other in a big slipstream circles. When this happens you can stop moving and descend to the ground and take the enemy by surprise from below. There are more tips I could give, but I suggest you get tips somewhere else regarding sheevsshee battles.
That's basically what I have to give about banshees. I could say more, but I have said the bare bones of shee tactics in multiplayer. Hopefully you will become an ace shee pilot, and you will make everybody angry but unable to complain because you got them without running them over like a noob. I get this satisfaction frequently.