Attach a change event handler to your input
You're only initially hiding your next element. You should also recheck on every change to your input value. Try this instead:
$(function(){
// bind a change event handler
$('#myDiv').change(function(){
this.value.length && $('#next_btn').show() || $('#next_btn').hide();
}).change(); // set initial state
});
I haven't used if statement since you're doing simple one sentence stuff in each case. I've rather replaced it with a boolean expression. Boolean execution of the Javascript engine will ensure that only one jQuery selector will be executed, so there's also no need to cache next button element.
You can of course replace that one-liner with an if like so:
if (this.value.length)
{
$('#next_btn').show();
}
else
{
$('#next_btn').hide();
}
Don't forget to initialize state
Attaching a change event isn't everything. You need to set initial state as well when the page loads (or is being ready). My code does that by the last call to .change(). This means that it first registers the handler and then invokes it as well.
Super simplified solution: using toggle with Boolean parameter
The whole thing can be replaced by this:
$(function(){
// bind a change event handler
$('#myDiv').change(function(){
$('#next_btn').toggle(!!this.value.length);
}).change(); // set initial state
});
Is change event ok?
It may not be that change event satisfies your requirements, because it fires after field looses focus. Maybe you should be using keypress or keyup events instead. But solution stays as is. Just replace event handler binding.