How do you make a YouTube channel private?
My goal is to make a channel of my baby videos, but so only people with the link can view it.
This way I can make sure only family sees the videos, but I won't have to send a link with every video.
You cannot make channels private. You can only disable them, which then hides them – but the videos become unavailable then, too.
To achieve your goal, I recommend uploading your videos as unlisted and then setting up a private or unlisted website (perhaps on sites.google.com) to collect your videos.
Note that if someone shares the link of an unlisted video/website publicly, the content becomes essentially public. So even if you upload videos as unlisted, make sure you don't include stuff that can be used against you.
To make sure that only people you trust can view the video, you have to upload them as private and manually enter the email addresses of anyone who is supposed to view them.
I believe the answer above is incomplete. You CAN make VIDEOS you post 'PRIVATE'.
The difference between PRIVATE and UNLISTED is that PRIVATE videos can only be seen by those you 'Share' them with, and those people must have a YouTube account themselves. (Very simple to create if they already have any Google accounts, such as gmail.)
UNLISTED videos will NOT show up in search results, but anyone who has the URL (which you can share with anyone you want, whether or not they have a YouTube account) can see them.
A good article on this, updated as of June 2018, can be found at http://smallbusiness.chron.com/build-private-youtube-channel-59230.html