why don't you try fixing your circuit arrangement (moving the relay), changing the 10M resistor to a much lower value, and then using that to get an actual idea of how the circuit is supposed to work, and then put something together and just trying it out?
Your measurement of 10M ohms seems high, unless you are using distilled/deionized water, and/or a terrible electrode setup, in which case your simple transistor setup is probably not going to cut it anyway... A pair of electrodes very close together (or better, overlapped finger pairs on a PCB) would probably be much better.