The responce that climbing stairs expends more energy than descending is correct. However, the question is not fully answeredd with out looking at concentric contractions (of muscle fibers) and eccentric contractions. Concentric activity is when the brain tells a muscle to contract and the muscle shortens (think of the ‘press’ in a bench press) it usually works agains gravity and is represented by climbing the stairs. Now when you go down stairs your muscles still contract with each step, but the muscle bodies lengthen (think of the ‘negative rep’ in said bench press). As any body builder will attest to the negative rep is where all the action is. Eccentric contrations carry the greatest risk of muscle injury and correspondingly can generate the most force.
So going down stairs may be physically more exerting due to the eccentric nature of the movement. Compounding this is that people usually decend at a faster rate than they climb. Muscle fiber recruitment must be more immediate in order to generate the requiate power to controll the decent. Think of plyometric exercises (box jumps, rope jumping, or medicine ball workouts) for examples of how exausting these simple movements can be.
Going up stairs uses more energy, but you get a better workout and are at a higher risk of injury on the way down. If yolu dont believe me go to your local guy, do a few sets of squats and then try climing and descending any old flight of stairs you find. Just be sure to hold the railing!