We see in the series that Jayne came over because of a slight increase of pay, and having his own room. Typically, people who turn for that kind of stuff are willing to turn again. Why has Jayne remained loyal for so long after making the decision to switch who he is working for?
Jayne did turn against them.
In the episode Ariel, Jayne betrayed the crew by calling an Alliance officer in an attempt to cash in on the rewards for Simon and River.
Unfortunately for Jayne, the Alliance officer double crosses him, and proceeds to arrest Jayne along with the Tams so that he can keep the reward money for himself.
Once Jayne realizes this, he helps them escape, and tries to pretend like their capture was accidental.
However, neither the Tams nor Mal were fooled.
Mal addressed this by knocking Jayne unconscious, and threatening to dump him out the airlock (pointing out that betraying his crew was the same as betraying him personally). Jayne appeared honestly repentant, and seemed to accept that Mal was, in fact, about to kill him.
As a result, Mal decided to spare him.
Simon Tam, on the other hand, waits and bides his time until Jayne is injured during the episode Trash.
Once Jayne is brought to the infirmary, Simon paralyzes Jayne, reveals that he knows that Jayne betrayed them, and points out the benefits of not turning against the guy who would be healing up someone in such a dangerous profession:
You're in a dangerous line of work, Jayne. Odds are you'll be under my knife again, often. So I want you to understand one thing very clearly: No matter what you do or say or plot, no matter how you come down on us, I will never, ever harm you. You're on this table, you're safe... 'cause I'm your medic. And however little we may like or trust each other, we're on the same crew. Got the same troubles, same enemies, and more than enough of both. Now, we could circle each other and growl, sleep with one eye open, but that thought wearies me. I don't care what you've done, I don't know what you're planning on doing, but I'm trusting you. I think you should do the same. 'Cause I don't see this working any other way.
Of course, Jayne being Jayne, it is questionable whether the "carrot" would keep him loyal should a sufficiently profitable (and apparently safe) opportunity present itself.
Therefore, River presents the "stick" as further motivation:
Also, I can kill you with my brain.
So between fear of Mal and River, concerns about the possibility of being helpless while Simon performs surgery on him, and, most importantly, the object lesson that Alliance officers can't be trusted, it is unlikely that Jayne would find an opportunity to betray the crew that he would consider safe, profitable and reliable enough to risk again. Perhaps he could find opportunities among the criminal elements he came from, but those are the same circles Mal is part of, and the deal Mal offers is apparently comparatively good to what other people looking for hired guns might offer.
Sure, Adelei_Niska might offer a pretty sweet deal, but Jayne is probably fully aware of just how much additional risk he'd take on working for Nishka. Plus Niska demands far more discipline than Mal does, and far more than Jayne seems capable of.
There are some other points in which Jayne would probably not betray the crew of the Serenity. I emphasize on probably, since Jayne is about as draconian as they come.
1) Pilot - As far as we know, Jayne cannot pilot a ship. Any act of betrayal inbetween points A and B will probably not end well for him. Unless he threatens Wash, if he wants...
2) Zoe - We know where her loyalties lie. Between Mal (former war comrade and superior) and Wash (husband and pilot), Zoe will always side with Mal and the ship. And Zoe isn't a push over by any means. And Jayne knows it. Not to mention...
3) Mal - A decorated war veteran himself, Mal is also a crack shot and a quick draw. While Jayne favors violence (sub guns, knives and grenades), Mal is always ready with that gun of his, unless Jayne gets a suckerpunch in.
4) Nowhere to run - Jayne isn't exactly personable. Where would he go? The Alliance probably has a small bounty on his head, and several other locations probably wouldn't mind picking him up. The Serenity is one of the few safe havens Jayne has...
5) Home - It's somewhat suggested that Jayne really has nowhere else to go. It's never mentioned about his origins, save that he never fought for the Independence (browncoats) or for the Alliance. So who does Jayne side with? Obviously, whoever is paying, but Jayne isn't exactly Boba Fett.
Jayne's ethics are pretty strong, just closely defined. At the start of the show, his loyalty is clear: Jayne is loyal to Mal alone. Hardly to the rest of the crew, and not at all to the Passengers. I assume the loyalty is to a man he considers to be a stronger one, as he pushes at everyone else at times but Mal.
First, he repeatedly expresses this to Mal in exactly this phrasing: "I wouldn't turn on you, Mal" Mal always has to restate it in terms of the entire crew. And later, the crew grows to include the Tams, which Jayne is even less likely to credit.
So on Ariel, he betrays the Tams, which Mal treats as a personal betrayal and sets up the kill scene. Jayne saves himself by telling Mal to lie about why he's dead - showing that he now considers his image to the crew as important, so they are now 'persons' in his eyes. So Mal relents and the issue is closed.
So by the end of the show, Jayne considers the whole crew as his team and won't betray any of them.