if you only have 1 element in your page follow this:
var searchField = document.getElementsByClassName("gsc-search-button")[0];
searchField.value = "GO";
Example : http://jsfiddle.net/uN2NV/
for multiple elements Try this:
var searchField = document.getElementsByClassName("gsc-search-button");
for(var i =0; i<searchField.length;i++)
{
searchField[i].value = "GO";
}
Example: http://jsfiddle.net/uN2NV/1/
For distinct element among multiple elements that share the same class:
var searchField = document.getElementById("distinct");
searchField.value = "GO";
Example: http://jsfiddle.net/uN2NV/2/