What is the basis for the Catholic practice of only the priest drinking from the common cup during communion? I've heard from a number of people that in most Catholic churches, only the priest drinks from the common cup in communion. Is this true?

Assuming it is true, what is the basis for this teaching?
Communion is always "valid under both species" as Catholics would say. That means that if you only receive Jesus under the form of wine or under the form of bread, you're receiving the whole of God. 

For about 20 years or so (depending where you live) in much of North America it has been common practice to offer the cup to the laity at communion for all masses on every day of the week. This is extra-ordinary, and not necessary and seems to be going away. The diocese of Phoenix was the first to go back to the old norms and my diocese (Madison, WI) just went back starting this past advent. 

The rationale out priest gave was that the bishop felt that absence would make the heart grow fonder. And on those occasions when we could recieve from the cup, we'd be more apt to do so.  When half the congregation just passes the Eucharistic minister by like there's nothing in the cup, you get to seeing why absence is necessary for a time. Hopefully, come Holy Thursday or Corpus Christi we can get a few more people considering Who they're passing up. 

The other good (well, I think it's good) thing the new reversion in norms does is limit the excruciating number of Eucharistic ministers (I mean extra ordinary ministers) any time you have to have a dispensation to recieve an indult you're looking at an institution within the church that won't last.  Nothing doctrinal about it, the Church is free to let lay people distribute communion and recieve in both species as well as end that practice. 

If there were enough priests and deacons to distribute communion in the first place the practice might have continued or even spread, I'd imagine.