How can I bind to the change event of a textarea in jQuery?

Try this actually:

$('#textareaID').bind('input propertychange', function() {

      $("#yourBtnID").hide();

      if(this.value.length){
        $("#yourBtnID").show();
      }
});

DEMO

That works for any changes you make, typing, cutting, pasting.

Use an input event.

var button = $("#buttonId");
$("#textareaID").on('input',function(e){
  if(e.target.value === ''){
    // Textarea has no value
    button.hide();
  } else {
    // Textarea has a value
    button.show();
  }
});

This question needed a more up-to-date answer, with sources. This is what actually works (though you don't have to take my word for it):

// Storing this jQuery object outside of the event callback 
// prevents jQuery from having to search the DOM for it again
// every time an event is fired.
var $myButton = $("#buttonID")

// input           :: for all modern browsers [1]
// selectionchange :: for IE9 [2]
// propertychange  :: for <IE9 [3]
$('#textareaID').on('input selectionchange propertychange', function() {

  // This is the correct way to enable/disabled a button in jQuery [4]
  $myButton.prop('disabled', this.value.length === 0)

}

1: https://developer.mozilla.org/en-US/docs/Web/Events/input#Browser_compatibility
2: oninput in IE9 doesn't fire when we hit BACKSPACE / DEL / do CUT
3: https://msdn.microsoft.com/en-us/library/ms536956(v=vs.85).aspx
4: http://api.jquery.com/prop/#prop-propertyName-function

BUT, for a more global solution that you can use throughout your project, I recommend using the textchange jQuery plugin to gain a new, cross-browser compatible textchange event. It was developed by the same person who implemented the equivalent onChange event for Facebook's ReactJS, which they use for nearly their entire website. And I think it's safe to say, if it's a robust enough solution for Facebook, it's probably robust enough for you. :-)

UPDATE: If you happen to need features like drag and drop support in Internet Explorer, you may instead want to check out pandell's more recently updated fork of jquery-splendid-textchange.

bind is deprecated. Use on:

$("#textarea").on('change keyup paste', function() {
    // your code here
});

Note: The code above will fire multiple times, once for each matching trigger-type. To handle that, do something like this:

var oldVal = "";
$("#textarea").on("change keyup paste", function() {
    var currentVal = $(this).val();
    if(currentVal == oldVal) {
        return; //check to prevent multiple simultaneous triggers
    }

    oldVal = currentVal;
    //action to be performed on textarea changed
    alert("changed!");
});

jsFiddle Demo