For those who use Thunderbird as their email client and use IMAP instead of POP, you can set it to have all of your emails stored locally by default without clicking the offline mode. The trick is a couple of settings in the advanced config editor (Options->Advanced->Config Editor):
- mail.server.default.autosync_offline_stores to true (you might have to create this value if it doesn't already exist. Right Click->New->Boolean)
- use_status_for_biff to false
More information is here.