Turn on advanced permissions under Blogs Settings>Features.

Make your family members of the "members" group (which is only available after you turn on advanced permissions).

They will then be the only ones who can see the protected posts (after they log in).

Check their permissions Blogs Settings>User Perms and Blogs Settings>Group Perms

Turn on Email Subscriptions under Blogs>Settings>Features.

If you don't want comments, turn them off for the blog, therwise people will be notified by email of these as well.

Likewise turn off public access if the blog is for members only.