"어떻게 궁금해요?" 1. 어떻게 궁금해요? => How are you curious? / In what way are you curious? 2. 얼마나 궁금해요? => How curious are you? 3. 어떤 게 궁금해요? => What [which] are you curious about? #1 is very odd and almost never heard in real life, #2 may be heard but rare, while #3 is common.

"어떻게 궁금해요?" doesn't make sense. Isn't it "어떤게 궁금해요?" "어떤게 궁금해요?' means "What do you wanna know?"

"How curious (are you)?" I would say ;)