From what I gather from your use case is that you don't want to use checkboxes, but radio buttons. If that is the case, this would be a good way to implement what you want:
$('input[type=radio]').on('change', function() { //event on all radio buttons
if($(this).attr('id') == 'check_other' && $(this).prop('checked')) {
$('#check_input').fadeIn();
} else {
$('#check_input').fadeOut('fast');
}
});
If you do want checkboxes, you could change the code a bit and probably get what you want.