BJT base connected to ground and still operates?

Simply redraw the circuit and you will see from where this base current is coming from.

^{simulate this circuit – Schematic created using CircuitLab}

As you can see the \$Q_2\$ base current is coming from \$\textrm{Vee}\$ voltage source.

And flows this path:

+Vee--->R1---> Q2 base-emitter junction--->-Vee

And remember that in the electronic ground is just a reference point, not a real Earth ground. And we are measure all our voltage with respect to this reference point called ground.

To measure the voltage we need two points in the space. One of this point is treated as a reference point.

We have a very similar situation when we try to measure a height of an object. We need a reference point. The most common reference point is "above mean sea level".

But when you measure the height of the table in your house the floor now becomes your reference point.