Curiosity killed the cat.
But originally (sixteenth century, Shakespeare, Wikipedia, just go with it), the saying was "Care," meaning worry or concern, "killed the cat." Which seems like the opposite of how we mean curiosity. Curiosity is something innocent to the cat in our version of the saying, until it becomes persistent, and then it becomes dangerous.
Of course, curiosity can originate from anxiety instead of innocent interest. As in "Siri, I'm curious to know the shape of common melanomas." Curiosity born this way has an urgency to it, but finding an answer brings only relief. Excess knowledge gets shaved away like so many moles.
Certainly curiosity can induce anxiety in its object. "Why are you so curious?" This is especially the case when curiosity seems to have a secondary motive. Or when curiosity for innocent motives unearths something about which the object of curiosity feels sensitive.
Curiosity has other connotations that suggest strangeness, weirdness, something a bit off -- curiosity shops and curious fellows. The guy standing on the corner. The kid setting fire to ants with a magnifying glass. A bit odd. A bit portentous. But eliciting little more than a shrug, a "hmm." And isn't that weird? Because the easiest way to make the object of curiosity less curious is to ask, "Hey, Mac, why are you standing on the corner?" or "Hey kid, what's with the fire and the ants?"
Can curiosity be brave?
Maybe in the sense that it takes some guts to admit "I don't know." Maybe in the sense that admitting that one does not know something implies a kind of weakness, so that curiosity comes from the place of confessing that one has a gap. Then again, I know that others characterize a willingness to admit these kinds of gaps as, itself, a kind of strength.
Can curiosity be sexy?
In that curiosity entails passion and probing insight, sure. It has to be directed toward actual desire to know something or someone. Or, maybe, it has to be effectively sold as that genuine desire. Otherwise, it's just a series of interview questions as a socially acceptable form of conversational foreplay. In the sense that titillation always has to involve something un-obvious, it has to surprise. There's nothing sexy about "Why don't you like vodka?" (Check please, one thinks, eyeing the door and hoping that one can drink in peace later). Whereas, "What do you think it says about you that you don't like vodka?" is asking for thought.
Curiosity is an invitation to thought, to conversation, to connection.
Can curiosity be boring?
If the area of inquiry is small but not in the sense of specific. "How was your day?" is trivial, dull, the epitome of blank domesticity. It is less an invitation to answer a question than it is to merely speak. "What was the moment in your day when you felt most alone?" is haunting, maybe overly so for dinner conversation. If it is broad but not grand. "Tell me about yourself," means nothing. "Tell me what you think about the fact that the universe is expanding into nothing while you're down here just eating a slice of pizza and scrolling through Facebook" should mean almost too much.