They say that we can’t change our noses but we can change our Moses. In the context of your question I would say that to show interest in a positive manner is good, but to be “nosey” is not.
There is no sin of being “too nosey”, per se. However, nosiness tends to lead to a pattern of negative interpersonal behavior. This often involves behavior that Judaism categorically forbids, such as listening to and believing bad and harmful talk about other people (“lashon hara”) and spreading rumors that can embarrass or trouble another (“rechilut”).
I once saw a picture of a mouth with a line drawn to negate it posted on a friend’s fridge. I assumed it meant to watch one’s diet and eat healthy, but he told me it was really a reminder to guard our mouths from speaking in a negative way about people. After reading your question I think I might suggest he add a nose to the picture on the fridge.