Elite Dangerous cannot be used with Steam family sharing as account sharing is against our EULA.
The exception to this is if both players own their own copies of Elite Dangerous, but in this case then both players should be able to get their own Steam keys anyway so that they can play together.
If you have any other questions about Steam family sharing please see the Valve FAQ here: http://store.steampowered.com/promotion/familysharing
From the Valve family sharing FAQ:
Can all Steam games be shared with friends and family?
No, due to technical limitations, some Steam games may be unavailable for sharing. For example, titles that require an additional third-party key, account, or subscription in order to play cannot be shared between accounts.