Bootstrap 3 modal centrado

Estilos

.modal {
  text-align: center;
  padding: 0!important;
}

.modal:before {
  content: '';
  display: inline-block;
  height: 100%;
  vertical-align: middle;
  margin-right: -4px;
}

.modal-dialog {
  display: inline-block;
  text-align: left;
  vertical-align: middle;
}

JS

function setModalMaxHeight(element) {
  this.$element     = $(element);  
  this.$content     = this.$element.find('.modal-content');
  var borderWidth   = this.$content.outerHeight() - this.$content.innerHeight();
  var dialogMargin  = $(window).width() < 768 ? 20 : 60;
  var contentHeight = $(window).height() - (dialogMargin + borderWidth);
  var headerHeight  = this.$element.find('.modal-header').outerHeight() || 0;
  var footerHeight  = this.$element.find('.modal-footer').outerHeight() || 0;
  var maxHeight     = contentHeight - (headerHeight + footerHeight);

  this.$content.css({
      'overflow': 'hidden'
  });
  
  this.$element
    .find('.modal-body').css({
      'max-height': maxHeight,
      'overflow-y': 'auto'
  });
}

$('.modal').on('show.bs.modal', function() {
  $(this).show();
  setModalMaxHeight(this);
});

$(window).resize(function() {
  if ($('.modal.in').length != 0) {
    setModalMaxHeight($('.modal.in'));
  }
});

Fuente: https://codepen.io/dimbslmh/full/mKfCc