Take a look around line 2068 in your css and reduce padding for the widget area –
.header-image .site-header .widget-area {
padding: 40px 0;
}
It’s the widget area that is causing a lot of the extra padding/space that you see, so once you adjust this it will reduce the space for the entire header.
The 40px represents the top and the bottom, and the 0 represents the left and the right. So by changing the 40px
you’ll be adjusting the top and bottom evenly.