As Diodeus stated, IE doesn't allow anything but the default border for <select>
elements. However, I know of two hacks to achieve a similar effect :
Use a DIV that is placed absolutely at the same position as the dropdown and set it's borders. It will appear that the dropdown has a border.
Use a Javascript solution, for instance, the one provided here.
It may however prove to be too much effort, so you should evaluate if you really require the border.