Kleinstein summed it up well. A drain to source voltage is not needed for a MOSFET to be conductive, but you need some voltage to drive a sensible current through it. It's like a water faucet that can be open or closed or something in between but unless there is some water pressure on one side, water is not going to flow through it. MOSFETs also have something called a body diode that is in parallel with the drain to source and in an N-channel device, conducts when the drain is negative relative to the source. Perhaps this was what you were thinking? AFAIK, the voltage and/or current has no lower limit ie., no threshold - it may just be swamped by random noise or leakage current from various sources.
Conceptually, If current is flowing in all electrodes at the same time and you just want to measure one of the currents, there are a number of MUXs available to do what you want. Assuming the HV source is positive relative to ground, the selected channel will connect the electrode to the picoammeter. The unselected channels' voltages will rise until the protection diodes in the MUX inputs conduct and divert the current into the Vdd power supply. Providing the current is limited to under a few microamps, this should be well within the operating conditions of the MUX. However, during a "spark gap" event, the current can reach damaging levels. To protect the MUXs, fast, high pulse current, low leakage diodes have to be connected to the electode inputs to divert the high currents to a voltage a volt or two above ground. Series resistors should also be added between the electrode signals and the MUXs to limit the current into the MUXs during a spark gap event.
Some caveats about this scheme: the electrode voltage would be either essentially ground, which is the burden voltage of the picoammeter plus the voltage drops through the protection resistors and MUX on resistance or it will be what the electrode voltage will be clamped to by the protection circuitry. From the use of the term "high impedance" the impression I get is that the MUX is not only selecting the electrode current to be measured but also selecting which electrode is attracting the solvent. Sort of like an inkjet printer when the ink is selectively placed. This would imply that the MUX would have to be able to switch, or at least stand off 4 KV which I don't think is feasible with semiconductors, at least inexpensively.
I mentioned the thermionic devices because as far as I know, low leakage, high voltage stuff is done in a vacuum.
BTW, when I looked up Electrospray and Taylor Cones I was thinking they are one of those "Wow, would you look at that" effects.