Infinities can be tricky.
The force between two charged particles varies inversely with the square of the distance between them. The energy required to increase the distance between two oppositely-charged particles from d1 to d2 is the integral of the force over that path. Even if d2 is infinite, this integral has a finite value.
This result generalizes to large collections of charges on, say, the plates of a capacitor. What this means in terms of your question is that the capacitance of the two plates does not actually tend toward zero as they are moved apart, and the voltage does not go to infinity. One way to interpret this result is to say that each plate individually has some minimum value of capacitance to the universe "at large".
It may help to visualize this not as two parallel plates, but rather as two concentric spheres, and allow the outer sphere to grow to infinite radius.
It may also help to draw the analogy with gravity, which is another inverse-squared force. An object falling to the surface of the Earth, even from infinitely far away, has a finite amount of energy (and a finite velocity) when it arrives.