Do devotions to Mary have any biblical foundation?

​Strictly speaking, the Bible does not explicitly say, “Pray the rosary and other devotions to Mary.” But, of course, that’s a somewhat obvious answer. The real answer to your question has to do with the biblical foundation of those practices.  
 
There are numerous ways that devotion to Mary is found in the Bible. Here are some of those ways.  
 
First, the Rosary, which is the most common devotion to our Blessed Mother, is primarily Scriptural. On each bead of the rosary, we pray the words of the Angel Gabriel to Mary, as well as the words of St. Elizabeth to Mary. “Hail, full of grace, the Lord is with you.” And, “Blessed are you among women and blessed is the fruit of your womb” (Luke 1:26-46). Additionally, we pray the “Our Father” prayer, which is directly from Jesus, and the “Glory Be” prayer, which is echoed in Scripture many times. So, yes, the Rosary is biblical in that sense. We are primarily repeating passages from Scripture. Additionally, in praying the Rosary, we are to meditate upon one of twenty different Scriptural stories during each of the twenty decades. So the prayers are Scriptural and the meditations are Scriptural.  
 
Second, devotion to Mary in general comes to us from the fact that Jesus lived the Fourth Commandment perfectly. That Commandment says, “Honor your father and mother.” Jesus honored His mother perfectly, and we are to imitate Jesus in all He does. Therefore, we are to honor His mother, too.  
 
Third, at the Cross Jesus said to the Apostle John, “Behold, your mother.” And in giving Mary to John, the Church has always seen that Mary is given to all of us. We are all to behold our mother in the person of Mary.  
 
Fourth, Jesus gave Peter the Keys to the Kingdom of Heaven and said that whatever he binds on earth is bound in Heaven (Matthew 16:19). He gave Peter the gift of infallibly leading the Church into all truth. We believe that this gift of infallible teaching did not stop with Peter but was passed on through the ages to all his successors, the popes. The popes have definitively taught, through countless ages, the value of honoring our Blessed Mother. So though this last point is not a direct biblical call to honor our Blessed Mother, it is an indirect biblical command through the teaching of the Church founded by Christ in the Scriptures.  
 
It would require a book to detail all the ways in which Mary is spoken in the Old and New Testaments. Hopefully, the thoughts above help.