Common Base Amplifier
If the base is grounded then the transistor is turned off, so how does
  the input reach the output?


The collector, base, and emitter currents are identical in the two circuits below.  If the transistor in the left circuit is biased on, the transistor in the right circuit is biased on.

Yet, in the left circuit, the emitter is grounded while, in the right circuit, the base is grounded.

So, I think you should carefully reconsider your assumptions.  Note that, in the common-base circuit, the emitter is no longer grounded.







simulate this circuit &ndash; Schematic created using CircuitLab