According to Canon 925 of the Code of Canon Law, Communion may be given under the form of bread only, wine only "in the case of necessity", or under both species "according to the norm of the liturgical laws". I don't have a copy of the Missale Romanum, which would contain the appropriate liturgical rubrics, but everywhere I've gone in the United States and Ireland it's been typical to offer both species.
EDIT: I've found this copy of an older translation of the 1566 catechism, which was the one in force up until the current 1983 catechism was published. That catechism has this to say:
Why The Celebrant Alone Receives Under Both Species
It is clear that the Church was influenced by numerous and most cogent reasons, not only to approve, but also to confirm by authority of its decree, the general practice of communicating under one species. In the first place, the greatest caution was necessary to avoid spilling the blood of the Lord on the ground, a thing that seemed not easily to be avoided, if the chalice were administered in a large assemblage of the people.
In the next place, whereas the Holy Eucharist ought to be in readiness for the sick, it was very much to be apprehended, were the species of wine to remain long unconsumed, that it might turn acid.
Besides, there are many who cannot at all bear the taste or even the smell of wine. Lest, therefore, what is intended for the spiritual health should prove hurtful to the health of the body, it has been most prudently provided by the Church that it should be administered to the people under the species of bread only.
We may also further observe that in many countries wine is extremely scarce; nor can it, moreover, be brought from elsewhere without incurring very heavy expenses and encountering very tedious and difficult journeys.
Finally, a most important reason was the necessity of opposing the heresy of those who denied that Christ, whole and entire, is contained under either species, and asserted that the body is contained under the species of bread without the blood, and the blood under the species of wine without the body. In order, therefore, to place more clearly before the eyes of all the truth of the Catholic faith, Communion under one kind, that is, under the species of bread, was most wisely introduced.
There are also other reasons, collected by those who have treated on this subject, and which, if it shall appear necessary, can be brought forward by pastors.
On the other hand, there's this from Aquinas' Summa Theologica (Third part, Question 80, Article 12 - "Is it lawful to receive the body without the blood?")
I answer that, Two points should be observed regarding the use of this sacrament, one on the part of the sacrament, the other on the part of the recipients; on the part of the sacrament it is proper for both the body and the blood to be received, since the perfection of the sacrament lies in both, and consequently, since it is the priest's duty both to consecrate and finish the sacrament, he ought on no account to receive Christ's body without the blood.
But on the part of the recipient the greatest reverence and caution are called for, lest anything happen which is unworthy of so great a mystery. Now this could especially happen in receiving the blood, for, if incautiously handled, it might easily be spilt. And because the multitude of the Christian people increased, in which there are old, young, and children, some of whom have not enough discretion to observe due caution in using this sacrament, on that account it is a prudent custom in some churches for the blood not to be offered to the reception of the people, but to be received by the priest alone.
Compare the current catechism, para. 1390:
1390 Since Christ is sacramentally present under each of the species, communion under the species of bread alone makes it possible to receive all the fruit of Eucharistic grace. For pastoral reasons this manner of receiving communion has been legitimately established as the most common form in the Latin rite. But "the sign of communion is more complete when given under both kinds, since in that form the sign of the Eucharistic meal appears more clearly." This is the usual form of receiving communion in the Eastern rites.
It appears, therefore, that there was not a Biblical basis for the older practice of Communion under only one species, but largely a group of closely related practical and theological considerations.
Sorry for the lengthy answer, but I wanted to get everything I could find on the question.