These are the styles which hide the border around the select box:
Code:
div.infs-widget-view-changer span select {
border: medium none;
font-size: 11px;
}
Just change it to "border: medium solid" and it should get the border displayed again.
By the way: there is something wrong in how "X items per page" selectbox displayed either - it shows the list bullet for the item and displays the box lower than the list of available pages.