someClassInstance.constructor.name is exactly the correct way to do this. Transpilers may not support this, but it is the standard way per the specification. (The name property of functions declared via ClassDeclaration productions is set in 14.5.15, step 6.)

As @Domenic says, use someClassInstance.constructor.name. @Esteban mentions in the comments that

someClassInstance.constructor is a Function. All Functions have a name property...

Thus, to access the class name statically, do the following (this works with my Babel version BTW. According to the comments on @Domenic your mileage may vary).

class SomeClass {
  constructor() {}
}

var someClassInstance = new SomeClass();
someClassInstance.constructor.name;      // === 'SomeClass'
SomeClass.name                           // === 'SomeClass'

Update

Babel was fine, but uglify/minification did end up causing me problems. I am making a game, and am creating a hash of pooled Sprite resources (where the key is the function name). After minification, every function/class was named t. This kills the hash. I am using Gulp in this project, and after reading the gulp-uglify docs I discovered there is a parameter to prevent this local variable/function name mangling from happening. So, in my gulpfile I changed

.pipe($.uglify()) to .pipe($.uglify({ mangle: false }))

There is a trade-off of performance vs readability here. Not mangling the names will result in a (slightly) larger build file (more network resources) and potentially slower code execution (citation needed - may be BS). On the other hand, if I kept it the same I would have to manually define getClassName on every ES6 class - at a static and instance level. No thanks!

Update

After the discussion in the comments, it seems like avoiding the .name convention in favor of defining those functions is a good paradigm. It only takes a few lines of code, and will allow full minification and generality of your code (if used in a library). So I guess I change my mind and will manually define getClassName on my classes. Thanks @estus!. Getter/Setters are usually a good idea compared to direct variable access anyways, especially in a client based application.

class SomeClass {
  constructor() {}
  static getClassName(){ return 'SomeClass'; }
  getClassName(){ return SomeClass.getClassName(); }
}
var someClassInstance = new SomeClass();
someClassInstance.constructor.getClassName();      // === 'SomeClass' (static fn)
someClassInstance.getClassName();                  // === 'SomeClass' (instance fn)
SomeClass.getClassName()                           // === 'SomeClass' (static fn)

Getting class name directly from class

Previous answers explained that someClassInstance.constructor.name works just fine, but if you need to programmatically convert class name into a string and don't want to create an instance just for that, remember that:

typeof YourClass === "function"

And, since every function has a name property, another nice way to get a string with your class name is to just do:

YourClass.name

What follows is a good example of why this is useful.

Loading web components

As the MDN documentation teaches us, this is how you load a web component:

customElements.define("your-component", YourComponent);

Where YourComponent is a class extending from HTMLElement. Since it is considered good practice to name your component's class after the component tag itself, it would be nice to write a helper function that all your components could use to register themselves. So here's is that function:

function registerComponent(componentClass) {
    const componentName = upperCamelCaseToSnakeCase(componentClass.name);
    customElements.define(componentName, componentClass);
}

So all you need to do is:

registerComponent(YourComponent);

Which is nice because it's less error-prone than writing the component tag yourself. To wrap it up, this is the upperCamelCaseToSnakeCase() function:

// converts `YourString` into `your-string`
function upperCamelCaseToSnakeCase(value) {
    return value
        // first char to lower case
        .replace(/^([A-Z])/, $1 => $1.toLowerCase())
        // following upper chars get preceded with a dash
        .replace(/([A-Z])/g, $1 => "-" + $1.toLowerCase());
}