Take a look at the picture below from my website: www.kokorugs.com
I am using Boostrap and I believe that there are some conflicting CSS styles.
The problem
The second border you are seeing is in fact not a border but a box shadow. It is beeing added by your bootstrap css to all inputs, but not desired for the search box. You should turn it off by countering it for the search box only. Add something like this to your css:
.gsc-input-box input[type="text"] {
-webkit-box-shadow: none;
box-shadow: none;
}