How to store Curry Leaves for many days?
Put almost all your herbs and leaves in the freezer if you want to keep them for more than 2-3 days. Either in a tight sealed container or ziplock bags. Keep them dry (eg. between layers of kitchen paper as suggested in the comments) so it's easy to separate them when you want to take out some and leave the rest.

Some herbs like cilantro or softer basils will loose their structure when frozen so they are not suitable as raw garnish after freezing (Suggestion in the comments).