The previous answer is somewhat incomplete.
While the Catholic church and most Protestant churches do not have any rules about menstruation and ritual purity, this concept is quite common in Orthodox churches, notably both the Russian and Greek Orthodox churches.
There, a menstruating woman is not allowed to participate in some aspects of liturgical life:
When I entered a convent of the Russian Orthodox Church Abroad in France, I was introduced to the restrictions imposed on a nun when she has her [monthly] period. Although she was allowed to go to church and pray, she was not to go to Communion; she could not kiss the icons or touch the Antidoron; she could not help bake prosphoras or handle them, nor could she help clean the church; she could not even light the lampada or iconlamp that hung before the icons in her own cell […] we simply presumed that menstruation was a form of “impurity,” and we had to stay away from things holy so as not to somehow defile them.
This quote is taken from an article by an Orthodox nun that criticises this practice. This article also describes the history of this prohibition at length. Since you are interested in the scriptural basis of this prohibition, I will not describe its church history in detail. The relevant part is that in early Christianity, some groups stayed close to Jewish spirituality, and hence were influenced by the Levitical rules around menstruation. Other Christian groups such as those in Antioch explicitly tried to differentiate themselves from the Jewish population and forbade their members from following the Levitical laws:
" …You shall not separate those who have their period, for even the woman with the issue of blood was not reprimanded when she touched the edge of our Savior’s garment; she was rather deemed worthy to receive forgiveness of all her sins." (from the Didaskalia)
The scriptural reference here is to an episode that appears in three of the gospels, where a woman who has been suffering from twelve years of bleeding is cured when she touches Jesus’ cloak. When Jesus notices her, he blesses her and is unconcerned about the ritual impurity that would normally affect a Jew who touched a bleeding woman. This is consistent with many texts in the new testament that emphasise the importance of faith and love over ritual laws.
However, some church authorities have read the story of the bleeding woman and concluded that her reluctance to approach Jesus should be the model for Christian women, who should therefore not enter church while they are menstruating. Therefore, arguments for banning menstruating women from liturgical life have co-existed with arguments encouraging them to participate since the earliest centuries of Christianity. In Orthodox churches, the former have generally won out.
Historically, menstruating women have also sometimes been banned from Catholic churches. However, this was much less common, and evidence of such prohibitions in the Catholic church cannot generally be found after the 17th century. The 6th century pope of Rome, St. Gregory, had this opinion:
A woman should not be forbidden to go to church. After all, she suffers this involuntarily. […] She is also not to be forbidden to receive Holy Communion at this time. If, however, a woman does not dare to receive, for great trepidation, she should be praised. But if she does receive she should not be judged. […] The menstrual period is no sin; it is, in fact, a purely natural process. But the fact that nature is thus disturbed, that it appears stained even against human will – this is the result of a sin…So if a pious woman reflects upon these things and wishes not to approach communion, she is to be praised. But again, if she wants to live religiously and receive communion out of love, one should not stop her.
This hints at another understanding, that the woman’s menstruation is a penance for the sins of Adam and Eve, and that therefore, like any penance, she may not want to receive communion while it’s going on. But there is no mention of ritual purity, and certainly no ban.
In conclusion, while some Christian churches do consider menstruating women impure, the majority of Christians do not. Even in Orthodox churches, the understanding is slowly changing, and many congregations, especially in western countries, no longer ban menstruating women from liturgical life. The scriptural basis consists of three aspects:
- The Levitical law, which some Christians try to follow in some respects, certainly considers menstruating women to be ritually unclean.
- The story of Jesus healing a bleeding woman has ben interpreted both in favour of and in opposition to menstruating women’s participation in the liturgy.
- There are references throughout the New Testament that declare that Christians have once and forever been purified by their baptism, and hence can not become unclean by violating the Mosaic law or other circumstances.
All quotes in this answer stem from this article: Ritual Impurity
Here’s a response to the article, giving additional (non-scriptural) reasons for the prohibition: On "Ritual Impurity": In Response to Sister Vassa (Larin)
And here’s an article focusing on the “impurity” of women after giving birth, that goes into the theological implications: Purify Her Uncleanness