How does a common base transistor amplify current if a small part of its emitter current gets divided into base current and the rest goes to collector current? If the input signal is applied to the emitter part of the transistor then the output current in the collector part should be less than the input one. Then how is the current amplified where the current flow actually decreases rather than increasing?
A common base transistor doesn't amplify current (meaning its current gain is less than unity), for reasons you have noted. It is often used as a current buffer, with a current gain of nearly one. A current source might have a low output resistance, which causes the current delivered to its load to drop if the load resistance is not much less than the source output resistance. A current buffer can provide a high output resistance so that the source current can be delivered to loads with much higher resistance.
The common base (and common gate) configurations are essential components in the very common cascode amplifier.