Which transistor should I use in my circuit?
Here is a bit better way: 







simulate this circuit &ndash; Schematic created using CircuitLab

If the probe resistance drops below some tens of M ohms, the beeper will sound. No more than 9uA flows through the probe so electrolysis is minimal. This will work with beepers up to at least a couple hundred mA, above that and Q2 may get too hot while partly on for a TO-92, so you could use a TIP31 or similar TO-220 part (just for Q1). 

C1 is to keep the battery voltage more steady as the battery ages (under varying load during the audio frequency beeping cycle), C2 prevents RFI from triggering the beeper. R4 and R2 prevent leakage from triggering the beepr and control the maximum resistance that will trigger the beeper. R1 limits the probe current to prevent electrolysis or overdriving the base of Q1. 

Q1 and Q2 are in a Darlington configuration which provides a theoretical current gain of perhaps 50,000, which is tamed somewhat by R2/R4.