A good solid question. First, let's expand the question a bit: there are three classes of people in this play: the rude mechanicals, the nobles—and the fairies. Why does this matter? Because when Titania and Oberon are first introduced, we learn they've been feuding, and that the upset and upheaval in all the rest of the land comes in part from this upset. That's part of the reason, perhaps the only reason, why the lovers are so star-crossed, and they can't work things out until the fairies do.

Now, turning to your specific question, Shakespeare was less showing a split between the classes than commenting on love. There are no women in the rude mechanicals, and once a woman is introduced, and Bottom becomes an ass, literally. It is as if he (Shakespeare) is saying that men alone might be fools (as the mechanicals are at the start of the play), but they are bigger fools when women are involved.  (Also, the lower classes are often shown as more innocent in Shakespearean plays, as if their lack of responsibility for ruling freed them somewhat.)