I've been working with es6 for two years now and never even heard of this happening to someone. Given the huge number of people who use babeljs, if there were to be a bug in it, it'd be reported and fixed rather quickly.
Yes, transpiling es6 -> es5 is adding another layer of complexity to your codebase, another thing that could potentially go wrong, but the reality is that the development gains we get from being able to use es6 are far more important than a small chance of a problem. It works well enough that you don't even think about it after leaving webpack in watch mode.
Block scoping (using const/let instead of var which is scoped to the function) alone is really nice, but I also make heavy use of destructuring, arrow functions, template literals, export/import syntax and promises. Es6 makes javascript more pleasant to code with, and can greatly improve readability.
es6 has become pretty well supported by modern browsers, so maybe after a year or so, we'll just be calling it "javascript" and running it without babel. I expect everyone to continue to transpile, though, to get newer features as they come, as the plan for ES is to release new language updates on a yearly basis.
It adds some extra nice language features to JS. It really is a good productivity improvement, especially with JSX which enables HTML literals: pretty much essential when working with a complex React front-end in my opinion.
My team has been using it for most of this year now and we've had no problems so far.
Yes, transpiling es6 -> es5 is adding another layer of complexity to your codebase, another thing that could potentially go wrong, but the reality is that the development gains we get from being able to use es6 are far more important than a small chance of a problem. It works well enough that you don't even think about it after leaving webpack in watch mode.