Carbonic acid in water If we add a weak acid such as carbonic acid in water it weakly dissociates to form bicarbonate and hydrogen ions . Now water already has some hydrogen ions, having a ph=7.  

So now indirectly we added hydrogen ions and base (bicarbonate ) ions to water which already has a hydrogen ion concentration.  The resulting solution is acidic . This means that in the addition of equimolar hydrogen ions and bicarbonate ions( base) hydrogen ions dominate.  

The only reason which I can think of is this: lets say there were 1000 hydrogen ions in water, now addition of 100 hydrogen ions and 100 bicarbonate ions will cause a net increase in hydrogen ion concentration because bicarbonate wont be able to counteract the increase in hydrogen ions because bicarbonate, when acting as a base, will be in equilibrium with carbonic acid. I.e. 100 bicarbonate ions won be able to neutralize 100 hydrogen ions present in the solution, and because of their equilibrium they will ionize only say 50 and overall, and there will be an increase of 50 hydrogen ions.  

Is my explanation correct?
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.