Hey,

Great question.

You are correct, (polar) protic solvents hydrogen bond to halides and (polar) aprotic solvents cannot.

I remember this because protic solvents have protons which are the polarized hydrogens you need to hydrogen bond.

In aprotic solvents, halide nucleophilicity follows basicity rules.

I remember this because a is close to b, and in aprotic solvent your halides are ruled by Acid Base. (This one is kinda a hokey stretch)

Please let me know if this unclear and you need clarification.