It is not possible to password-protect a link. It is also not possible to password protect a Flickr Set. You could use a Guest Pass to share private photos with people but if the photos are made private, they would not be displayed in the gallery.
One possible solution would be to provide a link to a web page in a password-protected directory.
Once the correct password is entered, the web page would simply redirect the user's browser to the Flickr Set page.
The actual Flickr Set itself would not be password-protected but a user would have to hunt it down manually to find it.
The link within your gallery would essentially require a password in order for the Flickr Set page to be displayed via clicks alone.
The web page you would link to would look something like this:
<!DOCTYPE html>
<html>
<head>
<meta http-equiv="refresh" content="0; url=https://www.flickr.com/photos/12345678@N01/sets/12345678901234567/" />
<title></title>
</head>
<body>
</body>
</html>
You would put the web page in a directory of its own on your web server and password-protect the directory using .htaccess and .htpasswd files.
These two web pages are a useful resource with instructions for setting up password-protection on a specific directory using .htaccess and .htpasswd.
http://www.javascriptkit.com/howto/htaccess3.shtml
http://www.tools.dynamicdrive.com/password/
I hope this helps.