Only regionally. In the global context, it's quite neutral.

People in Asia use "monkey" as a cutesy word for kids, to highlight that the kid is strong/healthy and can climb well.

Why do you think I mentioned that hackers are upvoting a comment calling people of a certain skin color monkeys?

I look forward to the day where I can call a black man a monkey and it only be considered an insult instead of a racist insult.