Carbonic acid in water
To start off, here is the equilibrium we are talking about so we can refer to it later:  

$$\ce{H2CO3 &lt;--&gt; H+ + HCO3-}$$  

Since the pH of the solution is $\mathrm{-log[H+]}$, the pH of pure water will decrease upon the addition of carbonic acid.  

Where I think you are getting confused is regarding the action of the bicarbonate that is produced. As this is an equilibrium reaction, bicarbonate will indeed act as a base, which is the reaction from right to left above. But lets say a molecule of bicarbonate reacts with hydrogen ion to reform carbonic acid. That just means another molecule of carbonic acid will dissociate to take it's place. This is all just a way of saying that the reaction shown above is an equilibrium, such that both the left to right and right to left reactions are occurring. But in the end there is never any net hydroxide produced or hydrogen ions consumed by this reaction, as there is always plenty carbonic acid available to counteract the basic action of the bicarbonate.  

I hope I understood your question correctly. Feel free to use the comments below to ask for any clarifications.