Yes, the meter must have power supply inputs that can be connected to the power rails of your circuit (you may need to increase this voltage if you can't find a meter that will work from 6V).
The inputs to the meter need to be independent of the power supply connections, and they need to respond to the voltage difference between the two input connections, which will be connected to the ends of the shunt. Therefore the meter will respond to the voltage across the shunt. These types of inputs are called differential.
Other points if you're considering a digital display:
It may need to be quite sensitive - ones I've seen have a full scale of 200 mV which corresponds to a full scale of 200 uA base current assuming you're using a 1K shunt resistor. You can of course change the shunt resistor. You should probably show the shunt resistor in the reference schematic of your circuit, since it will have some effect on the circuit's behaviour, especially if its value is high.
Ideally it should be configurable for decimal point position and units displayed. You don't want it to show "1.23 V" when the quantity you're interested in is actually "123 uA". I don't know if you'll be able to find a meter this flexible. Perhaps you should look for a digital meter that is already pre-configured to measure low currents. It would have the shunt resistor built into it, and it would display the correct unit.
If this is a teaching aid, I don't see any good reason why it's so important to build in a meter to show the base current - other parameters, like the base voltage and collector current, are just as important to understanding the circuit. If you're not going to put meters in for them, why have one for the base current? The reason we suggested putting the shunt resistor on the board was just to simplify the connection of an external multimeter to the circuit. Maybe you're taking the idea too far?