You can set to follow or not follow detected online state and choose options here: 'Menu icon' > 'Options' > 'Options' > 'Advanced' > 'Network & Disc space' tab

Configure offline settings click on 'Offline...' button

uncheck 'Automatically follow detected online state' select an option you prefer eg: Offline

Send unsent messages when going online? : select option you prefer, I would suggest 'Ask me' as it gives you the opportunity to check before sending.

Download messages for offline use when going offline? Select option you prefer, but if you have imap mail accounts, then you would be best to choose 'yes'. click on all OK's

I am using Tbird version 24, which is a little different, but this answer pointed me to the right place and answered my question.

In version 24: [Menu] > Tools > Options > Advanced > Configure offline settings > "Ask me for online state at startup"

See attached image.

For me, this up-front choice is actually better than always offline or always online. My thanks to Toad-Hall. I never would have found it.

A correction to my previous posting:

[Menu] > Tools > Options > Advanced > Network & Disk Space > Configure offline settings > "Ask me for online state at startup"