According to This Old House, most of the time the only reinforcement squeaky stairs need is tightening up the fasteners that keep each part together. Stairs are built from dozens of individual parts joined together, and the squeaking could come from any of the places where the treads you actually step on meet the others. Check each part of the squeaky steps to learn where the problem area is.
This Old House states that there are three basic components of stairs: the treads, the risers, which are the vertical kick plates beneath each tread, and the stringers, which are the saw-toothed framework that holds the other pieces in place. With use, any wooden stairs will start to squeak as the pieces are loosened and begin to rub against one another and the nails and screws the hold them together. The squeaking is caused by the tread moving relative to the riser, a stringer or both. Stepping on each part of the stair can show what causes the loudest squeak, and thus where the problem is.
This Old House warns that problems with the stringers are slightly more difficult to deal with than problems with the riser, since they often are not visible. Still, the process is basically the same. Find the current attachment points and have a helper stand on either side of the place you plan to add a new attachment to to press the boards together. Using a starter hole, add a new screw to the stairs, then cover with wood putty.