What is the effect of basis set on charge transfer?
Mulliken charges are not representative of charge transfer, I would argue, in any case. Mostly because of their enormous basis set dependency, as Geoff pointed out.

In any case, several points are missing here; for instance whether the geometries were optimized with each method/basis set combination. This is not trivial.

Assuming a single static geometry (or set of geometries) for all single point calculations, if you go with an MP2 approach, the larger basis set will give you the best result. This is so by definition. In fact, you should always use a large basis set with MP2 because this method converges slowly and requires a large amount of virtual orbitals. In DFT you never know, but I would try to use modern basis sets, such as Ahlrich's (RIP) "def2tzvp" (naming conventions change, but you will find it). Such basis sets are optimized for DFT, and are generally diffuse enough (as a rule of thumb, you should always include diffuse functions if you have an anion in your system).

For your goal, several methodological problems arise. First, if you are indeed optimizing the geometry, B3LYP might be very bad. Dispersion corrections or range separation might help, but you can check the B3LYP-MP2 difference to see if anything suspicious arises. 

Secondly, defining charge transfer is not easy. It is not trivial in SAPT (if you want to use SAPT i'd recommend PSI4 dearly for speed, there's a paper by A. Misquitta about charge transfer in SAPT but its not trivial to implement), which is the most rigorous energy term decomposition scheme, and in principle free from intermonomer BSSE. 

Defining charge-transfer is somewhat easier in some flavours of the many available EDAs, as in the Kitauma-Morokuma scheme which I think is implemented in GAMESS. Real space EDAs, as the Interacting Quantum Atoms method, might be useful here as well. I think real space charges are far more convenient than Mulliken's in any case. You see, charge-transfer is a nice term but a very diffuse quantity.

Otherwise, you might be better off by checking some more available, well defined observable. For example, the dipole moment of the dimer will capture the charge separation. 

In any case, good luck with this project!