Well, a one and a three year old may not be "babies" any longer, but they aren't ready for any particular neatness unless it's in their genes! Pulling books off shelves is a time-honored, favorite toddler game!
So you might try two things (you may get much better ideas than these). First of all, you might put a low table or something (not a basket unless it's a flat-bottomed one) near where you all read. You can say to your children, "When we're done reading a book, we put it on the table to be put away." Maybe that will help keep the books from so much chance of damage.
Another thing is to invent a new game. Make it fun! It's called, "Putting the Books Away." You'll have to do most of the game yourself, especially at first, but make it a habitual game in which your children hand you the books and help you put them on the shelf or in the cubby. You could add music or other things to make the game even more fun.
Neatness and order is something children have to learn (unless, as I said, they just have the prediliction for it). Pulling and throwing is much more "active" and more fun at toddlers. So you'll have to bear with it as you teach your kids this new, fun idea of putting books in good places.
However, the steps you're taking toward giving your children an interest in reading are marvelous! If some books are damaged along the way in spite of everything, in the long run it's a small price to pay for bring up book lovers.