Is there any secure version of the old Joomla! 1.6.x, or the 1.7.x series available?
Short and long answer is no. You need to upgrade a 1.6 or 1.7 site to at least 2.5.x as that is the most secure version (although 2.5 is fast approaching end-of-life). If you have a compatible template, you might be able to upgrade to 3.3.
Any Joomla website and to extend this a bit, any kind of software, especially open source ones and those that are publicly exposed, should be up to date, to ensure maximum security.
Joomla's current maintainable versions are Joomla 2.5 and Joomla 3.3. Joomla 1.6 and 1.7 were short term releases and we could say they were like the path from version 1.5 to 2.5. They were not meant to stay for long, they are not maintained (= no security patches, no bug fixes, no extensions support), thus they can't be used any more on production.
The updates, apart of the fact that they may add new features, they deal with known vulnerabilities and security issues. Those vulnerabilities are well-known to the public, therefore an older version of the software can easily be compromised.
Another condition, we should have in mind, is that 3rd party extensions are not also maintained for such older versions, giving even more space to vulnerabilities and other possible incompatibilities with the server software.
Usually, upgrading from 1.6, 1.7 to 2.5 and 3.3 is kind an easy task. But keep in mind that this is dependent on the specific conditions of each website, the 3rd party extensions and the possible level of customization and
It is strongly recommended to not try to upgrade directly on the live website, and always keep a working backup of the website.