Does ankle taping prevent injury or make the ankles weaker?

Helps prevent injury. A study of high school basketball players in wisconsin recently published in the american journal of sports medicine shows a reduction in the incidence (about 50% for athletes with prior sprains and about 65% for those with no previous sprain) of ankle sprains among athletes wearing ankle braces compared to those without braces. Braces have been shown to be more effective and cheaper than tape.
See below. Ankle taping helps to support the ankle and decrease the risk of injury.