No, Jesus did not have blood brothers. Yes, Mary remained a virgin.
In today’s Gospel from the Mass (Mark 3:31-35), many people mistakenly think that the Scriptures refer to blood brothers of Jesus and that Mary bore children after the virgin birth of Jesus.
 And his mother and his brothers came; and standing outside they sent to him and called him.  And a crowd was sitting about him; and they said to him, “Your mother and your brothers are outside, asking for you.”  And he replied, “Who are my mother and my brothers?”  And looking around on those who sat about him, he said, “Here are my mother and my brothers!  Whoever does the will of God is my brother, and sister, and mother.”
Confusion about this passage is based on ignorance of ancient Jewish culture, the Hebrew and Greek languages and the Church Magisterium. Here is a summary from the Ignatius New Testament Catholic Study Bible by Scott Hahn and Curtis Mitch that echoes what the Church teaches (find it here: every family should have this excellent book):
- The NT often mentions Jesus’ brethren (Matt 13:55; Mark 3:31; Luke 8:19; John 2:12, 7:3; Acts 1:14; Gal 1:9).
- The Church maintains that Jesus’ Mother Mary, remained a Virgin throughout her life. These so-called “brethren” of Jesus are thus His relatives but not children of Mary.
- These brethren are never called the children of Mary, although Jesus Himself is (John 2:1; 19:25; Acts 1:14).
- Two names mentioned, James and Joseph, are sons of a different “Mary” in Matt 27:56 (Mark 15:40).
- It is unlikely that Jesus would entrust His Mother to the Apostle John at His Crucifixion if she had other natural sons to care for her (John 19:26-27).
- The word “brethren” (Greek: adelphoi) has a broader meaning than blood brothers. Since ancient Hebrew had no word for “cousin”, it was customary to use “brethren” in the Bible for relationship other than blood brothers.
Is should also be noted, that St. Joseph, a pious man of great faith who had been given supernatural messages from God, would never have considered having marital relations with the Virgin Mother of God, for Blessed Mary was the New Ark of the Covenant. Joseph was too devout to even consider such a desecration. We should also remember that Our Lady would also never have considered breaking her vow to a consecrated life (Augustine tells us Mary took a vow of chastity which she never broke), especially after being granted the miraculous birth in which she mysteriously remained a Virgin (CCC 496-507).
Don’t listen to those who would profane the Mother of God.