I would certainly like to get the best crafting materials, but everything I'm gathering right now is green or blue quality. You can use them to craft higher quality materials, but given the numbers required that doesn't seem feasible. Is there a way to influence the quality of crafting materials my character gathers?
Normally wou will get items in the color of the destroyed item. Green gives green, blue gives blue. If you got to the DZ to get purple or even yellow items, you will get those materials upon destruction (or by levelling them, as explained in the question)
Edit on personal experience: Purple items only give blue material. There is no purple material in the game (yet). But you can increase your chance for better material by using the passive "skill" scavenging.