Using some jQuery and CSS you can make a cross-browser custom select (dropdown) box.

<div class="select-wrapper">
    <select>
        <option selected="selected" value="1">Option 1</option>
        <option value="2">Option 2</option>
        <option value="3">Option 3</option>
        <option value="4">Option 4</option>
        <option value="5">Option 5</option>
    </select>
</div>

.select-wrapper {
  position: relative;
  width: 215px;
  height: 30px;
  padding: 0 25px 0 0;
  margin: 0;
  border: 1px solid #0D5995;
  overflow: hidden;
  background: url(select-arrow.png) no-repeat right center transparent;
}

select {
  position: absolute;
  left: 0;
  top: 0;
  opacity : 0;
  width: 240px;
  height: 30px;
  padding: 5px 0;
  border: none;
  background: transparent !important;
  -webkit-appearance: none;
}

.select-wrapper span {
  display: block;
  width: 210px;
  height: 30px;
  line-height: 30px;
  padding: 0 0 0 5px;
}

$(document).ready(function() {
  $('select').after('<span>' + $('option:selected', this).text() + '</span>');
  
  $('select').click(function(event) {
    $(this).siblings('span').remove();
    $(this).after('<span>' + $('option:selected', this).text() + '</span>');
  });
})

See the demo.