The main effect of the Columbian Exchange on the civilizations of the New World was to destroy them. The Columbian Exchange weakened American civilizations severely. Because these civilizations were so badly weakened, the Europeans were easily able to destroy them or at least conquer and subjugate them.
The main way in which the Columbian Exchange did this was through germs. The Europeans who came to the Americas brought with them germs for contagious diseases such as smallpox. These diseases were completely nonexistent in the New World so the people of the New World had no immunity to them. When faced with infectious diseases to which they had no resistance, the people of the New World died in tremendous numbers. Because so many of their people died, these civilizations were terribly weakened. Because they were so badly weakened, it was easy for people like the Spanish Conquistadors to come in and conquer them. These European conquerors either subjugated the people (as with the Spanish in Mexico and Peru) or destroyed them (as the English and then the Americans did in North America).
In this way, we can see that the Columbian Exchange had a disastrous effect on the civilizations of the New World.