This can only be set up on the server: MobileMe provided the facility and iWeb was set to enable it directly. On any other server you will need to set it up yourself.
Some ISPs, GoDaddy among them, provide this as an option in their Control Panel. If your ISP doesn't you may be able to set it up yourself, provided that they allow you to upload an .htaccess file - not all of them do. The method is a bit fiddly and is described here:
This site offers a tool for doing this:
I don't know anything about them or how reliable they are. Googling on 'password protect web site' will bring up some other pages.
As Roger has already stated, this password protection needs to be set up on your server directly with the loss of this function with the demise of MobileMe.
There is another alternative however and that would be to re-build/transfer your site to one of the Content Management Systems, such as Joomla, Drupal or WordPress.org if your web host offers this facility. Most good web hosts will allow you to install the CMS's on your server and some have one click installation available.
It is very easy to set up password protected parts of your site using a Content Management System such as Joomla, because it is linked to a database that you need to link to. If you install it manually on your host, then you need to set up the database too and then link together, but most now have one click install, where the database is set up for you automatically - not as much flexibility like this over a manual install, but you can set up protected areas of the site and allow memebers to join and set up parts of the site that only members can get to. You can also set up different grades of membership too, so some can see some parts and others not.
If your site is in need to a revamp then consider this, otherwise stick with iWeb and just set up password protection directly on your server.