Typically, most server software has configuration options that allow you to specify which protocol versions are permitted (i.e. ssl_protocols in nginx, SSLProtocol in apache). Sometimes, they also bring defaults that are stricter than what OpenSSL supports by default.

The difference between older OpenSSL versions shipped by distributions and the latest version is not that newer protocols aren't supported, but that they haven't completely removed support for older, insecure protocols. For example, SSLv2 was only disabled by default in OpenSSL 1.0.2g, which was released today. Meanwhile, a lot of server software (e.g. nginx) had it disabled by default for quite some time now.

This is mostly relevant if you have code that uses OpenSSL directly - but then it's probably not a good idea to rely on OpenSSL defaults anyway.

tl;dr Older OpenSSL versions are probably fine as long as your server software has good defaults.