If Anakin had never fallen, and the world had been good, would Ahsoka have been Luke and Leia's aunt or big sister?
Alright, well if you’re talking a world where a) the Republic didn’t fall and wasn’t quite so corrupt, b) the Jedi weren’t toxic, and c) Anakin didn’t fall, and because all of the above “the world had been good,” then yes, absolutely she would have been. In fact, Anakin and Padme would likely have had more kids too.
However, if you mean that Anakin never fell, therefore “the world had been good,” I would not even be able to agree with your premise. What I mean by this is that Anakin’s fall was not the catalyst of the Republic and the Jedi’s destruction, it was a by-product. Anakin made his choice, but he was a result of their apathy. They created the situation that Vader arose from. Even if Anakin had never fallen, the Republic still would have. It was basically inevitable. Ergo, the world would not have been “good.”
So…what happens if Anakin doesn’t fall? Well, it depends upon how that happens of course, but let’s go with the simple and classic he doesn’t go to Palpatine’s office that day. Order 66 still goes down, the Jedi lead by Windu fail, but Anakin doesn’t fall. I’m sure Palpatine had a contingency plan for that. His plans were always meticulous. If Anakin didn’t fall, he would be a severe danger to him. So, what does Palpatine do? He sends clones to capture Padme, of course! Will these clones succeed? I don’t know. That depends on how good the handmaidens and Padme are, and how many clones are sent after them etc. However, Palpatine probably sent in a lot anticipating all of that, and even Anakin’s possible presence, so Padme gets captured, and Anakin (who was trying to help get what few children he could out of the Temple was not there) couldn’t have prevented it even if it were feasible. So, Padme is captured and held over Anakin’s head as leverage. Anakin, much like Galen Erso, will realise that it is better for him to unwillingly serve Palpatine, than to have a willing servant, (like the inquisitors), serve Palpatine, and he wouldn’t want anything to happen to Padme or his kids anyway. Besides, if he doesn’t cooperate, not only will he not be able to protect Padme, his children could very well be raised to be Sith, so…he’d better cooperate if he wants to do what’s best. Twin Sith Skywalkers loyal to Palpatine would not be good. So, we still get a Darth Vader of a sort.
A huge operation will be done to rescue Padme, (after Obi-Wan tries to kill Anakin and realises Anakin is only pesudo-fallen, and they get into a massive argument about “this is why we don’t have attachment, but he agrees the main thing is to get Padme and the children to safety). The question is of course does Padme survive it? Or does she die by being shot down by a clone while fleeing captivity, and Luke and Leia get born via c-section when their mother is instantly killed. (Imagine Obi-Wan using his lightsaber to slice open her pregnant belly and pull them out.) Of course, this gruesome scenario, assumes the kids haven’t been born yet. The timeline of the rescue is up in the air, so it could be they were already born, but regardless Palpatine is going to anticipate a rescue operation. He also has the Force to guide him. He’s not going to make a clean rescue mission easy, so Padme’s death is quite probable. Palpatine probably knew that he’d be safer if she was dead. If she couldn’t be held over Anakin’s head, she could be another huge threat to him. Furthermore, if the kids have already been born, she’s that much more expendable. He probably told his men to shoot her if she ever tried to leave, but to leave the kids alive. (Besides, he’s petty and he doesn’t like her, or Anakin, so there!) There’ll be stormtroopers, and probably inquisitors everywhere. It’ll be a nightmare to pull it off without causalities of some kind.
Of course, whether Padme survives, or doesn’t survive the rescue, Anakin is no longer needed as Palpatine’s lap dog, so he’ll try to kill him immediately, or leave ASAP to take care of Luke and Leia provided their mother did indeed die. In either scenario, there is, of course, no way Ahsoka “Fulcrum” Tano is not looking out for them. Even if she weren’t so emotionally invested in the Skywalkers, looking out for Force-Sensitive kids is right up her alley. So, yes, she would still, definitely, be a part of their life. I think.