Canning jars are actually great for plant storage. When an herb has even just a tad bit to much moisture though it can cause a sour smell, and in some cases mold growth.
After you have harvested your herbs let the dry in a dark dry location for at least a few day up to a few weeks depending on the type of plant. General rule of thumb is a stem should not snap but should break a bit when bent. This can be the trickiest part, practice make perfect.
Once they have reached that stage put them into your jars. For the next few weeks burp the jars a couple times a day. Every couple days dump them out and let the herbs open up.
Each plant is different, the goal is to dry them slow to preserve all the goodness. Over dry herbs are still usable, but not moldy herbs, so err on the side of dry rather than damp. When you think they are dry enough stop burping the jars and store as usual.
If when you do open them you have the kinda sour not so good smell they were just a touch to damp at sealing.
Hope this helps!