Just to give a different point of view from all other answers so far:
However, I have to share this password with other members of the family, so they can also login to it. Is using WhatsApp or Facebook Messenger secret conversation an acceptable method for this?
No. It neither matters that your Netflix password is probably not so important; nor does it matter what Facebook says about its encryption.
The point is that you cannot know what they are doing internally. Do they encrypt every message with an additional master unlock key? Do they re-encrypt the messages along the way? I assume that neither Whatsapp nor Facebook (the apps) have been reverse engineered/peer reviewed. And even if they had been, they could be changed on every single update.
Reasons are plentiful. "We want to make your private conversations searchable - of COURSE only YOU can search through them...", "We want to make sure your private conversations stay around when you lose your phone" etc. etc. Storing your data is their prime business, after all.
You never know, you cannot really know, and it can change at any update. That should give you enough pause.
Why does it not matter that it's "only" Netflix? Because it leads to complacency. If you are used to just chatting your unimportant passwords around, you will sooner or later just don't care anymore and also send the important ones.
Are there better methods?
Sure. Public Key cryptography was invented to make it unnecessary to post passwords in the clear. Have your family create private/public keys, have them mail their public keys around in the clear, have them verify each other over the phone or when you meet in RL next time, and then you are all set to share any secrets you like, in the future.
Feasible? Probably not, with the current interest of the general public in security issues being as marginal as it is. But still - better.
Then there's Threema, which has been peer reviewed and is at least open source in the security department, where it counts. Of course, here it's also not trivial to make sure your messages are not also encrypted towards a master recipient, but at least you have some things to verify, with the source.