How did the crusades have a lasting legacy on Europe, the Near East and Asia?
This is a great question. Let me give a few perspectives.
First, as you know the crusades were a fight between Muslims and Christians over the Holy Lands. If we think about it for a moment from a contemporary perspective, this is still going on. There is still fighting over religion or to put it more broadly, the ideology of the West and Islam.
Second, in the fighting, one of the great cities of the world changed hands. In 1453, we see the sack of Constantinople. Now it is called Istanbul. More importantly, many of the intellectuals found their way over to the continent. Through this we would see the Renaissance and the Protestant Reformation.
This is most likely going to be a poor answer but the crusades are acually what began the long lasting hatred between certain Muslims and certain Christians. As you most likely know the current problems in the middle east have been based off of those same principles and that is one of the longer lasting legacys from the crusades. Also the crusades caused the fall of constantinople which was the most advanced civilization in the world at the time, after constantinople fell the rensisance began which was the birthplace of modern mankind and the time period in which the most knowledge was discoverd in over 1000 years. Thus it could be argued that the crusades directly caused the beggining of the modern world and modern technologies.