Are asymptotic lower bounds relevant to cryptography?
I'll try to give a partial answer, since I'm not fully aware of how this issue is considered by the entire crypto-community (maybe repost on crypto.SE?).

I would say there are two "kinds" of cryptographers: Theoretical and  Practical. I will not try to tell them apart (every practical-cryptographer is also a bit theoretician..) but I'll say that for theoretical cryptography - this issue doesn't really matter. For any security parameter, there will be an instance size that will provide that security level, and that is usually all that we care about.

Practical cryptographers care a lot about the issue you mention. For a given security parameter (say $2^{1024}$) cryptographers try to come up with the most efficient protocols, shaving the constant as much as possible. Look for instance for NIST's AES or SHA-3 competition. The algorithms were required to be both secure and efficient.
The issue here is that the notion of security is somewhat different from the "theoretical" one, and sometimes is not really asymptotic.

A concrete example I can think of is the discrete log or the DDH assumption (the security of many cryptographic schemes are based on these assumptions). We assume that for some group $G$ computing the log takes $O(|G|)$ time. (We can't be sure: it might be that $\text{P}=\text{NP}$ and this problem can be solved in $O(\log |G|)$). Cryptographers DO care about the actual time it takes to compute the log. More accurately, they care alot about special cases for which computing the log is easy. In many papers you will see the sentence "Let $G$ be a group in which DDH
is hard". See this survey for families of groups for which DDH is believed to be intractable (unless P=NP).