First, let's get rid of the transistor, and note that the unloaded base voltage is zero:

simulate this circuit – Schematic created using CircuitLab
Now let's connect the transistor, and note that the V.BE voltage is zero as well. We assume that the BE diode does not produce any thermal currents.

simulate this circuit
Since V.BE=0, the base current is nil, and the approximate emitter-collector current is zero. Thus the collector load resistor has no voltage drop - both its terminals are at the same potential:

simulate this circuit
Now you may ask: OK, that's an idealized circuit analyzed in a simplistic fashion. How about a real circuit?
We can replace the R1+R2 divider with its Thévenin equivalent, and use a high-valued shunt resistor to measure the collector current:

simulate this circuit
According to the CircuitLab simulator, the voltage across the 10MOhm resistor is about 5uV, so the current is 5uV/10MOhm=0.5pA.
The choice of the voltmeter equivalent series resistance is not arbitrary: that's the input resistance of a most digital multimeters. Thus, the multimeter alone acts like a parallel combination of RVM1 and VM1.
I've picked a random 2N3906 from my bin-of-transistors, and connected it with base shorted to emitter, and used a 9V battery as the voltage source. The measured voltage - and thus current - is on the same order as the simulation predicts (10uA, give-or-take), and is very sensitive to external disturbances.
After connecting the base resistor, there's no significant change in the collector current, although the current was at the edge of what I could measure without shielding.
Due to the below-picoampere currents involved, low-leakage measurement technique is essential. No probe wires should touch, even though they are insulated - PVC is not the best insulator for such measurements. Silicone or Teflon (PTFE) is much better. Even body movements can affect the readout, and the entire circuit up to the voltmeter and supply source should be shielded from air currents.
Ideally, the circuit should be in a metal box acting as a shield, plugged directly into the multimeter. The transistor and any other insulating surfaces in the box, e.g. the banana plug standoff, should be washed in IPA followed by a deionized water rinse. It doesn't take much conductance due to contamination to skew the results here.
I might re-do the measurements later after sticking the circuit into just such a box.