Open the wishlist's Share settings and switch its visibility to Invite only. From then on, only you and the people you invite can open it. Everyone else lands on a "No Access" page.
Steps
Open the wishlist you want to make private.
Go to its settings and open the Share tab.
Set the visibility to Invite only.
Save.
Good to know
Once a list is Invite only, a plain link stops working for other people. To let one specific person in, invite them by email. See "How to share a private wishlist".
New wishlists are Anyone with the link by default, so nothing is private until you change this setting.
Making a list private doesn't change who can edit it. Editing is handled separately, through email invites and access levels (Owner, Full Access, Contributor, Viewer). You don't have to make a wishlist private to add collaborators, and a public list can perfectly well have people helping manage it.
Want to go the other way? See "How to make a wishlist public".
Related
Who can see my wishlist?
How to share a private wishlist
How to make a wishlist public
What information is visible to guests?