What you are describing is known as the Photon Number Splitting Attack (PNS), described for the first time (I think) by Brassard, Lütkenhaus, Mor ans Sanders in this 1999 paper. Several countermeasures have been invented since (single photon sources, robust protocols, decoy states), but detailing them would stray away from of your question.
If one sends 2 or more photons, the enemy can store this photon in a quantum memory, and make the polarization measurement after the basis choice have been revealed. The enemy then knows whether to measure in the rectilinear or diagonal basis, and has all the information shared by Alice and Bob.
In 2001, Scarani, Acín, Ribordy and Gisin proposed a protocol (SARG04 pdf, wiki) which is robust against PNS attacks while using the same physical modulation than BB84. This protocol works by exchanging classical information which is more ambiguous than the basis. However, as soon as $K\ge3$ and the losses are over 50%, this protocol is subject to an unambiguous discrimination attack.