If this were an important doctrine, it seems there would be fairly clear reference to it in the Scriptures. However, the Bible is incredibly silent on it and even seems to indicate that Joseph had no union with here until Jesus was born.
Irrelevant. There are many things all Christians believe that are not directly spelled out in the bible, such as the Trinity, the hypo-static union, etc. Also, Catholics do not believe in a sola scriptura approach to doctrine.
[B]y implication that Joseph would need to be an eternal virgin as well since he was married to her.
There is no such need. Joseph may have been a widower, for instance. His virginity is never referenced in Catholic tradition.
What importance does the eternal virginity of Mary have? What does it matter? A what would have been so horrible about her and Joseph enjoying the intimacy God provided for a married couple?
It's not that it would have been horrible -- indeed, it is good for a man to know his wife in this sense. In Catholic teaching, however, there are two other things at play here: the notion of sacrifice, and piety with respect to the holy. When we sacrifice things, we sacrifice good things. This goes for burnt offerings and little penances alike. When, for instance, a monk or a nun take vows of poverty and chastity, it isn't because sex and wealth aren't good things -- indeed, they are! But the spiritual good is better than the temporal good, and they are choosing, out of love of God, to give certain things up to seek further spiritual nourishment.
Piety and reverence to the holy is something that in the Catholic/Orthodox do a bit different than other traditions. For instance, traditionally the vessels of consecration, the tabernacle, the altar, etc. are all veiled. Women, traditionally, veil their head in prayer, particularly in the presence of the blessed sacrament. The veiling hides them from plain sight, not because they are bad, but because they are holy and beautiful. Joseph abstained from relations with Mary because she was the tabernacle -- she contained Jesus within her. She was the ultimate sacred vessel -- the Theotokos -- who bore God. Out of reverence, awe, respect, and love, he had forgone relations with her.
In addition to all this, it is a crucial article of faith that all Catholics are bound to hold under penalty of mortal sin. From the Second Council of Constantinople:
II. If anyone does not confess that God the Word was twice begotten, the first before all time from the Father, non- temporal and bodiless, the other in the last days when he came down from the heavens and was incarnate by the holy, glorious, God-bearer, ever-virgin Mary, and born of her, let him be anathema.