[For symmetric encryption quantum computers would only matter if they were pretty fast/cheap and we didn't have ready-to-go 256-bit symmetric crypto, but we do]
OpenSSH actually has an implementation of a reasonable contender for SSH. Google have experimented (in Chrome builds) with some of these contenders for TLS too. What you would likely want to do - since by definition these are relatively untried algorithms and so might be unsafe against adversaries with an existing not-at-all-quantum computer - is combine with an existing algorithm, most likely elliptic curve based, but RSA would be possible, under a "swiss cheese" model where you're only dead if an adversary penetrates all the layers.
But like I said, much worse. Given that there aren't any suitably large quantum computers (and it's always possible that we'll eventually figure out we just can't build usefully large quantum computers, just like we eventually found out that while you can travel faster than sound you can't travel faster than light) it would make no sense to deploy this today, even though it continues to make sense to do research Just In Case.
Slower, in a different complexity class from the classical ones (AKA O(n^2) instead of O(n), but I don't know what exact complexities the top candidates have right now, as a function of key length).
Larger messages, in that a signature requires 100's or 10's of Mb instead of a few kb.
Larger public keys, as again 100's of Mb instead of a few kb.
Larger private keys, as in a few to 10's of Mb instead of 1/2 to a few kb.
But I guess the most relevant "much worse" is that people have much less confidence that those algorithms are secure, and they are complex enough to hide huge traps that make implement RSA look like a walk in the park.
larger messages
larger public keys
larger private keys
