What's the best way to handle multiple sites on a shared server?

You have multiple options:

1. Use Rel Canonical on your pages, this ensures Google will never index a page on the wrong URL and Domain, I personally favor this out of all options.
2. Use a redirect when a user or bot visits the URL and then forces then back to where they should be, this can be done using htaccess or the httpd.conf
3. Upgrade your hosting.
4. Separate your hosting.

Rel Canonical

Simply using <link rel="canonical" href="http://www.example.com/pagename"/> on your URLS will prevent duplicates entering into Google, Very simple but very effective.

.htaccess

Using a htaccess such as:

RewriteEngine On
RewriteCond %{HTTP_HOST} !^www\.wrongdomain\.com/subfolder/
RewriteRule .* http://www.right-domain.com/%{REQUEST_URI} [R=301,L]

Upgrade your Hosting

Alternatively you could always upgrade your hosting to a WHM Cpanel or Plesk account that makes new users rather than new sites meaning that you get an extra layer of protection. This basically jails each site and malware, viruses and exploits are harder to leak though to the other sites. From a security stand point, this would be your best option. Most Hosts provide this but don't advertise it.. worth checking out.

Separate your hosting

Finally, if upgrading your hosting is something your interested in then it might be worth also considering using a 2nd host, nowadays its so cheap to find a host that only costs a few bucks a month and having 2 web hosts isn't a bad thing in fact many people believe it more solid for creating a back link to your other site (though some argue about C class, and not going there but ya..).