Prevent NPN Transistor Base-Emitter Current
current flows from the base to the emitter of the transistor which is
  still on.
  
  How do I filter out the smaller current when just the base-emitter on
  B is conducting, so the circuit functions like an AND gate?


You should understand the emitter current equals the base current plus the collector current (Ie = Ic + Ib). If you do not want the base current ("smaller current") then place the load on the collector side of the top transistor.







simulate this circuit &ndash; Schematic created using CircuitLab