Depending on the design of the bookcase, it may be possible to "rotate" the shelves. You can either swap the shelves around or you can actually turn them over. This may be the quickest way to make your home library look ok if, for instance, you are expecting company and have no time to purchase a new bookcase (which you should probably do eventually). While swapping shelves around and overturning them you may also consider putting thin durable plastic underneath the shelves. The plastic will not be noticeable, but it could stop further sagging.
It can also be a lot cheaper to replace just the shelves with more durable wood or plastic. You can even go with durable glass if it is designed to withstand heavy loads. This may greatly improve the look and originality of design in your home library.