Very, very basically: the advantages far outweighed the disadvantage of having to pay a dowry.

The most obvious advantage is the perpetuation of the class. If everyone goes "this dowry thing is stupid, I'm not playing," then none of the sons can get married, and then what happens to the nobility?

More structurally(?), the dowry had a purpose beyond enriching the family a woman married into. Early in the Middle Ages, it was common for grooms to pay a brideprice to the families they were marrying into, either explicitly or implicitly to their wives. By the high Middle Ages, primogeniture had set in, with its focus on keeping a family's wealth concentrated in the hands of only one descendant rather than splitting it between all children, and it became common for European aristocrats to endow their daughters with their inheritances early - as the dowry. (Or, if she wasn't to be married, to give to the abbey where she became a nun.) Seeing this as a payment to the other family is simplistic. In reality, it typically went to the new couple and therefore contributed to the daughter's maintenance. So there you have another advantage: you're making sure that your daughter has something to live on in her adult life, particularly important if she marries a younger son, because that primogeniture means that younger sons inherit very little land or money.

Then, there were benefits if the marriage broke down or ended. In the case of divorce, or if a court determined that the husband was too financially irresponsible, the dowry could be returned to the family or turned over to the wife. And typically a widow inherited her dowry (or its equivalent) from her husband when he died, though it might be earmarked for one or another of their children on her death. (This was something preferred by noble fathers and husbands: they could add legal riders to the conveyances to prevent pious women from giving their dowries to the Church of their own volition instead of keeping it in the family.)

Something else you've got to keep in mind - possibly the most important aspect of this answer, due to stereotypes of a) women being powerless pawns and b) valued only for their looks - is that it's not true that women were supposed to abandon their birth family and consider themselves solely a part of the one they married into. The fact is, women were informal ambassadors, from the young Englishwoman of a good family who married a powerful English magnate's son to the French princess who married the Holy Roman Emperor. They generally maintained robust social networks through letter-writing and patronage, and had the ability to wield a substantial amount of power this way. It was very much to a family's benefit to have their daughters married, and to offer as large a dowry as possible in order to get their feminine ambassadors into as influential a position as possible. English queens that came from the domestic nobility are fantastic examples - Elizabeth Woodville (married to Edward IV) and Anne Boleyn (married to Henry VIII) were in a position to recommend their male relatives to be appointed to lucrative posts/given titles and to help their female relatives to other good marriages.