Okay, I have seen little of the anime and this question has a few obvious answers (to me).
Let's start with Meowth:
Meowth has always wanted to belong. First, he learned to speak and walk on two feet just to impress Meowsy. Then Meowsy was a b!%&$ and took him as a freak for this. He was then kicked out of his 'pack' of Meowth because of his strange new ways. He has never belonged anywhere and only wants the boss to like him as a Meowth and not a stuck up Persian. So, he's always trying to impress the boss. Ever since the boss said he wanted that Pikachu, he's been determined to get it. Though it seems his determination goes down throughout the years.
Jesse was an amazing Team Rocket member, then she was paired with James and Meowth. They dragged her down and brought out her true, nutty personality. She probably just wants to be as good a Team Rocket member as her mother, whom disappeared while searching for Mew. So, like Meowth, she has been determined to get Pikachu ever since they found that this Pikachu was special and insanely powerful (though the power has also gone down over the years).
James was the rich kid who, oddly enough, didn't like his life. He didn't like being pampered and all of the other things that come with money. If I remember correctly, he wanted to get away from his parents because they didn't understand him, and had made him an arranged marriage. So, he joined Team Rocket to get away from them. He too wishes to impress the boss (though the reason escapes me) and he wants to stick with Jesse and Meowth.
The reason they are not dead is, the show would be nothing without them and you don't just kill off characters (or at least not all of them) in a kids show. Though, if you would like to see them get serious and stop with the blasting off, maybe you should watch Pokemon: Black and White (yes, sadly it is not Best Wishes in english)