i definitely can't speak to the technical aspects, but i do think his statement makes some naive assumptions about the security of analog communication. humans have been espionage-ing analog communication for way longer than we have digital, and we're good at it.
"Analog communication" covers things like radio transmissions, which are indeed no better than the Internet for the transmission of sensitive information. But with a written message carried by a trusted, experienced courier, it's not that difficult to determine if it might have been intercepted. For instance, the message could be kept in a briefcase with tamper-evident locks.
for another thing, though, neither the intelligence community, nor any other organization as large as the dnc, can anymore operate using analog communication. there's just too much data, and a significant amount of that data can't very easily be communicated in writing.
if we're just talking about using analog messages to send the very most important messages, like cipher keys in your example, then what you're saying makes a ton of sense to me; but, my understanding is that the dnc hacks were allegedly done through phishing links/social engineering/whatever, and i dunno that couriers would solve that (except to the extent that a single courier isn't going to carry 30,000 emails, but then we're back to the practicality of it). i feel like the solution is better training for the humans using the computers, not getting rid of the computers, so to speak.
I'm going to assume that by "analog communication" you mean "written messages", which can't easily be classified as analog or digital (you could argue it either way, using the pattern of ink or individual letters as your base unit).
No, couriers would not solve social engineering directly, but that doesn't negate the fact that they do have security advantages over computer networks. Whether or not those advantages would have prevented the DNC hacks is of no consequence to the validity of Trump's comment.
i'm also mildly skeptical that detecting a compromised courier network is easier than detecting as compromised digital network. not saying you're wrong, just that intuitively it's easy for me to imagine ways of compromising a person without leaving an identifiable trace or physical clue; i would think it's comparatively difficult to compromise a digital network without leaving a clue. the computer side of that is literally just speculation on my part, but i think you're underestimating how good people are at spying on people.
The problem is that the Internet is actually composed of many networks (that's what the word "Internet" means; it's an abbreviation of "inter-network"). When you send a message using the Internet, you are not only trusting your own network and your recipient's network (which you should have a strong guarantee of security for), but also every network in between, typically involving multiple ISPs and long-distance carriers.
Furthermore, the Internet was not designed to be secure. It was designed in an age when its only users were large academic, research and government institutions with the funding to purchase expensive mainframe hardware. There were no malicious actors, and everyone could be trusted to do the right thing. All security mechanisms on the Internet were built on top of it years later, once commodity hardware became commonplace, but the base infrastructure is fundamentally insecure to this day.
For example, occasionally
routing misconfiguration at an ISP causes network traffic to traverse a different path than it would have ordinarily. This means that you cannot even trust that the
same networks between you and your recipient are used for each message, or even each packet within a message. You cannot even trust that using the same IP address will deliver your message to the same host.
To use an analogy for courier transport, imagine you give your message to a courier for secure delivery to the Russian government. He leaves your secure government building and stops at a street corner and asks which way the Kremlin is. He then blindly follows in the direction pointed until he reaches the next street corner and asks for directions again. Eventually, he will come across someone and ask them for directions to the Kremlin, and they will reply: "This is the Kremlin, please give me your message." He hands it over and that's the end of it. That's how Internet routing works.
By contrast, a real courier would be able to recognise whether he has actually reached the Kremlin, or even whether he is in the right country. He would also be able to verify identity documents of the person he hands the message to as necessary. Most importantly, he would be able to provide a reasonably firm guarantee that nobody intercepted his message en route.
While it is
possible to use end-to-end encryption to securely send messages using the Internet, that still requires a known trusted key for your communication partner, which requires some other method of communication beforehand. (I'm ignoring the X509 infrastructure commonly used to issue SSL certificates here, because they require you to trust a number of corporations in order to establish a trust chain, which is undesirable for important government messages.)