Is there a term that describes a person's quality of being curious but getting into trouble because of it? For example, Odysseus in "The Odyssey" wants to see the cyclops so he decides to stay in the cave but because if it his crew almost dies. I am looking for a word that I can use to say: "Because of Odysseus's [insert word] he makes several decisions that nearly lead to the loss of his crew".
Curiosity itself can get one into trouble but it is more likely that a "morbid curiosity" leads to trouble. So you can use "morbid curiosity" in your example.
A morbid curiosity exemplifies addictive curiosity. It has as its object death, violence, or any other event that may cause harm physically or emotionally, the addictive emotion being explainable by meta-emotions exercising pressure on the spontaneous curiosity itself.
The metaphor "curiosity killed the cat" can support your subject also.
Consider prying, this sense of which is defined by oxforddictionaries.com as
Excessively interested in a person's private affairs; too inquisitive.
Another similar sense might be conveyed by nosiness, though this has an informal feel to it.
Mischievous is the word that you might be thinking of.