The standard noisy approach is not to try to determine the presence of an eavesdropper as such, but to create a final key where, even if there is an eavesdropper, you can still be confident that the eavesdropper has negligible information about the key. So you aren't trying to distinguish between noise and eavesdropping, but pessimistically assuming that all noise may be due to eavesdropping, and obtaining a key that is still private, even given this (or deciding that the noise level is too high for this to be possible).
This is done through privacy amplification - details of specific protocols (like Cascade) are covered in links others have posted, but essentially these are classical hashing protocols: classically, if you have a sequence of bits the eavesdropper has some partial information about, you can hash those down to a shorter sequence that the eavesdropper provably has negligible information about.
Proving the security of this approach for BB84 (including how much key, if any, can be obtained given a certain noise level) is more complicated, since the eavesdropper isn't limited to just knowing some fraction of the classical bits, they can do quantum attacks like entangling their own qubits with the ones in transmission. The Shor-Preskill proof of BB84's security (using some earlier work by Lo and Chau) works by showing that the protocol (with hashing) is equivalent to distillation of EPR entangled states from noisy entangled pairs, with EPR pairs being inherently uncorrelated (in a quantum or classical sense) with an eavesdropper.