Hi Alistair,
It's not a straightforward process.
Prerequisite:
Can you check if your customer is on version 6.1.0? if yes then you need to upgrade to build 8192 at least to use TLS1.1
Current status:
Adobe Campaign technical support can provide you the stack trace and in stack trace look for protocol versions, whether it's TLS v1.0 or TLS v1.1? If yes, you can simply change the MX management rules and it will work for you.
Implementation:
If you have access to serverConf.xml configuration file, then you can check the configurations if it's which is applied. If not configure it by installing the proper SSL certificate and allowing following configurations on the server.
SSLProtocol: all -SSLv3 -TLSv1 -TLSv1.1
Docs for TLS and SSL:
http://httpd.apache.org/docs/2.2/mod/mod_ssl.html#sslprotocol
Security/Server Side TLS - MozillaWiki
or simply raise a neolane support ticket to enable TLS 1.1 and if possible TLS1.2 as in future everyone is moving towards 1.2(at the moment not possible with Adobe campaign??)
Regards,
Amit