Step 1: Concept
According to Gauss's Law, the net electric flux $\Phi$ through any closed surface is equal to the net charge $Q$ enclosed divided by the permittivity $\epsilon_0$.
Step 2: Analysis
As the balloon is blown up, its radius increases, but the total amount of charge on its surface remains the same. Since the charge enclosed by the Gaussian surface (the balloon itself) does not change, the flux remains constant.
Step 3: Conclusion
The flux remains unchanged.
Final Answer: (C)