Powerset generation using recursion and time complexity
You should have converted you Java code into pseudo code.

In fact, if your code runs exactly as you describe, namely given a set $S$ as a string it generate all subsets of the symbols of the input string, then the running time is trivial. You just need to count the number of different subsets generated by your algorithm/code. Input size is the length of the input string, i.e., number of characters. So, if the number of characters is $n$ then there are total $2^n$ different subsets. So, the time complexity is $O(2^n)$.