The usefulness of quantum communication is in key exchange, but also in the way that it makes any attempt to eavesdrop on a communication channel obvious, because the eavesdropped will collapse the quantum states being transmitted. It doesn't matter if the observer can interpret the contents of the channel (and P=NP won't help them there) because their interference cannot be hidden. note though that this can constitute a denial of service making it impossible to send information across the channel, but that's not the same as knowing what is being sent
P=NP would render the key-exchange process pointless for all currently known symmetric ciphers (because 3SAT is NP-complete) and probably all our current asymmetric ciphers too (though it isn't clear at this point if eg. the discrete logarithm problem is in NP), but one can still use a quantum communication channel to securely distribute a one-time pad, safe in the knowledge that it cannot have been observed by a third party without your noticing (and if it was observed, throw it away and start again). P=NP doesn't crack one-time pads, so secure communication is still possible.
All that remains is to find a suitable encryption algorithm in a higher complexity class. We don't have one yet, but I suspect it is only a matter of time.