We get this question fairly often. The answer is...it depends. If you wear a rigid ankle brace that doesn't allow the ankle to move naturally, it can change your ankle mechanics by prohibiting natural movement. Your ankle area becomes out of balance, and in time, you may end up with a host of new problems caused by the rigid brace. The new problems could even go right up the body, so you may end up with knee, hip, or spine issues when you had just an ankle problem to start!
If you wear a soft ankle brace that adds support but still allows natural movement, it won't usually weaken the ankle. In this case, the bigger question might actually be: "Should I wear an ankle brace for added support at all?" Again, the answer depends on the situation. A soft ankle brace can be beneficial in preventing recurring injuries in athletes who have had previous sprains or "loose ankles" from rolling the ankle too many times. Using the brace solely as a preventative measure is not usually recommended, though. If you've been able to perform your sport and stay active without injury, that's ideal. It would be rare for a doctor to recommend adding a medical device like a brace into your routine in that case.
A board certified orthopedic doctor specializing in the foot and ankle area is the best choice to properly diagnose and treat ankle problems. It's a good idea to seek medical treatment after a problem occurs if it doesn't resolve itself in a reasonable amount of time with R.I.C.E.: rest, ice, compression, and elevation. Acute pain and recurring ankle issues also usually need medical attention.