First up, I've got to say that I'm unaware of anyone using these over 
HTTPS in production, so issues are forward-looking and largely hypothetical.

The good news is that both use DNSSEC:

http://dnssec-debugger.verisignlabs.com/hdl.handle.net
http://dnssec-debugger.verisignlabs.com/dx.doi.org

The bad news is that some servers in the dx.doi.org DNS rotation don't 
appear be listening on 443 at all and that those that do have variable 
configuration that gets them a 'C':

https://www.ssllabs.com/ssltest/analyze.html?d=dx.doi.org

Further, a number of doi.org-native links redirect from HTTPS to HTTP 
without warning. For example https://dx.doi.org/ links to 
https://dx.doi.org/help.html but that's just a redirect to 
http://www.doi.org/factsheets/DOIProxy.html www.doi.org isn't listening 
on port 443.

Testing DOI resolution over HTTPS gives occasional very long timeouts 
(presumably those non-443 servers?).

All of the servers in the hdl.handle.net  DNS rotation are listening on 
443, but again the variable security config and low scores:

https://www.ssllabs.com/ssltest/analyze.html?d=hdl.handle.net

Note that some of the servers have 'test' in their server name, which 
makes me wonder...

Again, the home site and help pages are HTTP only and there are HTTPS-> 
HTTP redirects.

Testing handle resolution over HTTPS seemed to work reliably for me when 
I tested it.

Anyone have ideas as to who needs to lobby who to get this improved?

cheers
stuart