Is there a written tradition that would explain why John the Evangelist is often depicted in art as effeminate?

Is there a written tradition that would explain why John the Evangelist is often depicted in art as effeminate?

There seems to be no known written tradition to indicate this. In art, St. John is generally depicted in one of two ways: the young beardless type is early (as in a 4th-century sarcophagus from Rome), and this type came to be preferred (though not exclusively) in the medieval West. In the Byzantine world the evangelist is portrayed as old, with long white beard and hair.

The legends that contributed most to medieval iconography are mainly derived from the apocryphal Acts of John. These Acts are also the source of the notion that John became a disciple as a very young man. Iconographically, the young beardless type is early (as in a 4th-century sarcophagus from Rome), and this type came to be preferred (though not exclusively) in the medieval West. In the Byzantine world the evangelist is portrayed as old, with long white beard and hair, usually carrying his Gospel. - St. John the Apostle

Early Christian art usually represents St. John with an eagle, symbolizing the heights to which he rises in the first chapter of his Gospel. The chalice as symbolic of St. John, which, according to some authorities, was not adopted until the thirteenth century, is sometimes interpreted with reference to the Last Supper, again as connected with the legend according to which St. John was handed a cup of poisoned wine, from which, at his blessing, the poison rose in the shape of a serpent. Perhaps the most natural explanation is to be found in the words of Christ to John and James "My chalice indeed you shall drink" (Matthew 20:23). - St. John in Christian art

St. John at the Crucifixion of Jesus in a Stabat Mater by Pietro Perugino, c. 1482.

Russian Orthodox icon of the Apostle and Evangelist John the Theologian, 18th century (Iconostasis from the Church of the Transfiguration, Kizhi Monastery).

St. John is often depicted beardless because he is believed to be the youngest of the apostles. It is simply a traditional way of expressing that. As for other representative ways of depicting St. John, I will let Wikipedia have a go at it:

John the Apostle c. 6 AD – c. 100 AD) or Saint John the Beloved was one of the Twelve Apostles of Jesus according to the New Testament. Generally listed as the youngest apostle, he was the son of Zebedee and Salome. His brother was James, who was another of the Twelve Apostles. The Church Fathers identify him as John the Evangelist, John of Patmos, John the Elder, and the Beloved Disciple, and testify that he outlived the remaining apostles and was the only one to die of natural causes, although modern scholars are divided on the veracity of these claims.

In Art

As he was traditionally identified with the beloved apostle, the evangelist, and the author of the Revelation and several Epistles, John played an extremely prominent role in art from the early Christian period onward. He is traditionally depicted in one of two distinct ways: either as an aged man with a white or gray beard, or alternatively as a beardless youth. The first way of depicting him was more common in Byzantine art, where it was possibly influenced by antique depictions of Socrates; the second was more common in the art of Medieval Western Europe, and can be dated back as far as 4th century Rome.

Legends from the Acts of John, an apocryphal text attributed to John, contributed much to Medieval iconography; it is the source of the idea that John became an apostle at a young age. One of John's familiar attributes is the chalice, often with a serpent emerging from it. This symbol is interpreted as a reference to a legend from the Acts of John, in which John was challenged to drink a cup of poison to demonstrate the power of his faith (the poison being symbolized by the serpent). Other common attributes include a book or scroll, in reference to the writings traditionally attributed to him, and an eagle, which is argued to symbolize the high-soaring, inspirational quality of these writings.

In Medieval and through to Renaissance works of painting, sculpture and literature, Saint John is often presented in an androgynous or feminized manner. Historians have related such portrayals to the circumstances of the believers for whom they were intended.[89] For instance, John's feminine features are argued to have helped to make him more relatable to women.bLikewise, Sarah McNamer argues that because of his status as an androgynous saint, John could function as an "image of a third or mixed gender" and "a crucial figure with whom to identify"bfor male believers who sought to cultivate an attitude of affective piety, a highly emotional style of devotion that, in late-medieval culture, was thought to be poorly compatible with masculinity.bAfter the Middle Ages, feminizing portrayals of Saint John continued to be made; a case in point is an etching by Jacques Bellange, shown to the right, described by art critic Richard Dorment as depicting "a softly androgynous creature with a corona of frizzy hair, small breasts like a teenage girl, and the round belly of a mature woman."

In the realm of popular media, this latter phenomenon was brought to notice in Dan Brown's novel The Da Vinci Code (2003), where one of the book's characters suggests that the feminine-looking person to Jesus' right in Leonardo da Vinci's The Last Supper is actually Mary Magdalene rather than St. John.

The Last Supper by Leonardo da Vinci.