How do you increase the level of or improve the quality of the rewards from Daily Craft Quests? Is it based on the level of your craft ability, or number of times you completed the quests etc.?
The level of the items from the reward containers depend on the location where you had to turn in the crafting quest. It doesn't matter where you started it.
The actual quest/target location you get depends on both, your own character level as well as your crafting line's main passive ability (the topmost one that defines which items you can produce).
For example, if you can't craft Voidsteel Gauntlets, you won't get the quest asking you to craft such items. So you also won't get the higher level reward containers including Voidsteel Ore.
The number of times you complete the quests doesn't have any influence on the actual rewards. These are really just counted for a few achievements.
As for the actual contents of the crafting reward containers, that's completely up to luck and you can't really change that. You might get refinement items being worth 5,000 gold or more, or you could get a single style material that's worth nothing.