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Hi,
How to apply the class to filter

You can locate filter element after grid initialization as
         grid.hdr.getElementsByTagName("SELECT")[0]
and assign any necessary css class for it. 

Or just customize dhtmlxgrid_filter.js , which contains HTML code
dhtmlXGridObject.prototype._in_header_select_filter=function(t,i){
      t.innerHTML="<select style='width:90%; font-size:8pt; font-family:Tahoma;'></select>";