The down arrows that appear on every drop down search field are just part of the <select> inputs and are not added by the S&F plugin.
To remove them would require some sort of hack. The answer to this can be found on this post on Stackoverflow.

A quick answer would be to add just this lines of code in your themes css file.

 .postform {
   -moz-appearance:window;
   background-repeat: no-repeat;
   background-position: right center;
}

Please read the post on Stackoverflow for further information!