I am using Botan utility to perform encryption. When I initialize my connection to a remote machine using SSH, I am able to trade keys over the secure SSH connection. However, sometimes I use inetd to establish the connection, and in this case, there is no security on the inetd connection, but I need to use it to trade keys with the remote machine.
I imagine there is some standard for this whereby I send a public key over an insecure channel and the remote end uses this to encrypt a key to send back to me over the insecure channel, which I can then decrypt to get the key.
What would be an example of this kind of protocol that Botan supports?
Without previous trust, or communication through a side channel, there's no way to do that. Diffie-Hellman kex allows you to establish a channel secure against others who don't participate in the connection, but you cannot verify that you're communicating with the intended recipient.
Classic MITM example: you connect to some remote endpoint, it receives your public key and sends you something signed with that key. However, you have no way to verify whether you've sent your key to the real destination, or whether the response comes from an attacker - therefore, you have a secure tunnel, but you have no information with whom you're securely communicating (the attacker may even connect to your intended destination and proxy the traffic, which passes over him unencrypted).
To be sure that you are indeed communicating with the intended endpoint, you need to exchange some sort of identification of the host beforehand or through a secure channel. SSH does this using the "fingerprints" - it asks you on first connection if you trust that host, and you're supposed to verify the fingerprint through an independent channel.