Many hundreds of years ago, daughters were considered property. So not only would Juliet have been betrothed to Paris, Paris would have paid a hefty dowry to the Capulets. It would have been worth quite a bit for Paris to marry into such a wealthy family, and if Juliet refused, then the entire dowry would have to be returned.
I would venture to say that Juliet's emotions and the plot events are more pertinent to her life. It is her marriage that is being arranged and her family members who are fighting and dying in the streets. Her life happens to her (with the exception of her decision to marry Romeo in secret) and Romeo makes life happen. The focus is more on the female whose life is full of twists and turns do to no fault of her own. She is expected to obey her father, other male relatives, and then finally, her husband. Being Catholic, however, having two husbands would not be acceptable. It is a series of problems and how to solve them in her household. The spotlight is on her because that's where the action is!
Although this question can be answered in any number of ways, I prefer to answer in a more controversial tone and say that the Capulet family is emphasized more than the Montague family because this story is, in essence, about love and not about politics. In regards to love, then, even in our own era, the parents of the bride are of far more importance than the parents of the groom. In Romeo's time, the subject of a dowry would certainly have been in question. This has everything to do with the wealth of the bride's parents. Ironically, this isn't even mentioned in Romeo and Juliet, and yet society still steered Shakespeare to focus more upon the Capulets. This, of course, is not to present the feud between the Montagues and the Capulets to be one-sided. The Montagues and the Capulets are both at fault and both responsible for the deaths of these star-crossed lovers, but at the end of the day, the play is about their love and not about their families.
It's to do with the structure of the play. Many critics have noticed that it's a play of two halves, but - as you rightly point out - it's not really a play about the Montagues and the Capulets. It is a play about Romeo and Juliet.
The first half of the play, I would argue, focusses mainly on Romeo: starting off with his relationship with Lord Montague (as much as you can describe it as a relationship!), moving through his friendships with Mercutio and Benvolio and his love for Rosaline, his being persuaded to go to the Capulet ball, and then a scene at the Capulet ball which focusses more on Romeo than it does on Juliet. Then, following the balcony scene, we see Romeo with Friar Laurence, Romeo with his friends (and then with the Nurse), before the pivotal fight scene which culminates with Romeo murdering Tybalt.
The second half of the play focusses more on Juliet and her family - where Romeo's family were (really) Benvolio and Mercutio - Juliet is, as a woman, trapped within her household. So this part of the play focusses far more on scenes within Capulet's household: Juliet's marriage to Paris is immediately on the agenda, we see all the preparations for her wedding, her falling out with Capulet, the wedding moved forward, her taking the potion in her bed, and then the wedding morning.
So you see how the play shifts its focus from Romeo to Juliet after the deaths of Tybalt and Mercutio. So why does Shakespeare not show us inside Romeo's household? Because Lord Montague and Mrs Montague, as Act 1, Scene 1 makes quite clear, are not Romeo's friends. To better illuminate Romeo, Shakespeare turns to his friends: the gang of Montague boys. They are - and I'm sure this is true for many teenage boys - his real family.
Of the two main characters in the play, who has more to lose? Young unmarried ladies of means in 14th c. Verona--much like young ladies of means in Shakespeare's time and in many places, like young ladies of means today--would have been committed to arranged marriages by their parents. Juliet's marriage to Paris--the Prince's cousin, no less--is politically advantageous to the Capulets, meaning the stakes involved in defying her parents' wishes are even higher than under normal circumstances, certainly higher than those Romeo takes, despite the fact that he risks his life for her. If Juliet is disowned by her family, she has nowhere to turn; if Romeo must, he can relocate and begin his life again. It's Juliet, not Romeo, who risks all in this story; her family's ambitions heighten her risk, so we must focus on these Capulets.