Floors that have become somewhat "springy" or are starting to sag toward the middle, while not uncommon, are not a normal part of your home's aging process, and should be fixed, not ignored. If the problem is on a first floor built over a basement, the problem can be fixed by installing a beam to support your joists, but if your house has crawl space or the problem is on an upper floor, a different approach is required.
The best way to stabilize floor joists in a crawl space (stabilizing upstairs floor joists will be discussed later) is by "sistering," or using new boards the same width and thickness (usually 2x8 or 2x10) as your existing joists to double the width of the joists.
Begin by cutting your board to the same length as an existing joist. Apply construction adhesive to the top edge, then brace it tightly into place against the existing joist and the bottom of the floor or subfloor using 2x4s. If there is sufficient room in your crawl space to allow hammering, attach the joists by putting in a pair of nails every 16 inches. Angle the first nail through the bottom of the new joist into the old joist. Install the second nail near the top of the joists and go straight through the side of the new joist into the old one. If the space is too tight, pre-drill holes and use screws and a drill driver to attach the joists. Repeat this process on all joists that are unobstructed by pipes or electric wires.
You may find that the space between some of your joists contains water and electrical lines, which makes sistering impossible. On joists where you can't add a sister board, you can still provide extra stability by installing a 2x4 to the bottom of the joist. Simply cut a 2x4 to the same length as the joist, apply construction adhesive to the 2x4, and brace it into place against the bottom of the joist. When the adhesive has cured (24 to 48 hours), put nails or screws (make sure to pre-drill) every 10 inches to permanently attach the 2x4 to the joist.
The process for stabilizing upstairs floor joists is the same as those in a crawl space, with one major difference--you need to go through the ceiling to access the joists. This makes it a much bigger project. If you decide to go forward, consider hiring a contractor to tear out and reinstall the ceiling or even do the entire project.