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There are three ways to farm Jute scraps:
I haven't found a place where Jute drops consistently like other crafting materials. Since Jute is such an integral part of the Tailoring Discipline, it's kind of worth your time as a tailor to discover a method for acquiring Jute quickly. Have you found a place to farm Jute quickly? Care to share? With farming not necessarily the best option, we turn instead to...
Salvaging was the best way to acquire Jute scraps prior to the Trading Post coming back online. You could salvage white or better weapons/armor and have a good shot at getting some Jute scraps. I found 90% of my scraps this way early on.
When salvaging, I would advise using crude salvage kits for white items and basic or higher salvage kits for blue or better. If you have an upgrade in the gear you are salvaging, be sure to use a basic or higher salvage kit in order to have a shot at recovering the upgrade.
I attempted to find items worth purchasing on the Trading Post for the purpose of salvaging, but in most cases I broke even or lost copper.
3. Trading Post
I would highly recommend selling crafting materials you don't need (like wood or ore) in order to buy more Jute scraps. This method appeared to be the fastest way to acquire Jute scraps while testing (that is, whenever the Trading Post was actually on).