Say I control Genesis Hydra, then an opponent casts Claustrophobia onto Genesis Hydra, do I control Claustrophobia?
You don't control Claustrophobia. Permanents enter the battlefield under the control of the controller of the spell, unless something else is explicitly stated.
For permanent spells:
608.3. If the object that’s resolving is a permanent spell, its resolution involves a single step (unless it’s an Aura). The spell card becomes a permanent and is put onto the battlefield under the control of the spell’s controller.
For other spells or abilities:
110.2a If an effect instructs a player to put an object onto the battlefield, that object enters the battlefield under that player’s control unless the effect states otherwise.
No, you will not control Claustrophobia. As with all permanents, Auras come into play under the control of the person who cast the spell that became the permanent. It doesn't matter who controls the permanent that the Aura is enchanting.
A permanent’s controller is whoever put it into play unless the spell or ability that put the permanent into play states otherwise. Other effects can later change a permanent’s controller.