Zelda's importance depends on the role the developers give her. There are a lot of games where Zelda wasn't present at all, so she certainly didn't have much importance in those games. Depending on which game she is in, her importance has varied. In The Wind Waker, she had a fairly large role, and I like that. Not only did she help the player at the end of your quest during the fight against Ganon, but she also helped throughout the entire quest and was a major character. Tetra was the leader of a band of pirates and, unknowingly, Princess Zelda the entire time. Though she may not always have a large role in a lot of the games, she usually does have a bit of importance.
In a lot of games her role was that of the generic princess. She gets captured and the hero has to save her, which is similar to what we see in the Mario series with Princess Peach. This was particularly present in the older generation of Zelda games, where they didn't give her much of a role at all except in a few simple tasks or a little information. She's still very important despite the smaller roles, however. I'm glad to see it deviating a bit from that path and making her a little more helpful in some of the games, though. The Wind Waker's Zelda was excellent, and I would like to see something like it again. Giving a character like Zelda a more important role makes for a more exciting game, in my opinion. She's only as important as the developers make her.
Over the years she has grown more important, varying from game to game, but still more important than she was in the past. I already mentioned how in The Wind Waker she had a huge role - though we didn't know she was Princess Zelda at the time - in helping Link get started with his quest. In Twilight Princess she helped in various parts of the game where it would have been impossible to continue without her, and it's those little things that make Princess Zelda have a very important role beside just being the objective in the games. She has been given increasingly more important roles as the games progress through time, and that's definitely a change for the better. She is a very important figure in the series, and I like to see her depicted as such.