The bar doesn’t have a fixed height but it does have a top/bottom padding of .5em. You can add some Custom CSS (Appearance > Customize > Additional CSS) to change the default padding. Example:
.easy-notification-bar {
padding-top: 4px;
padding-bottom: 4px;
}
Simply change the 4px to the value you want.
That said, the bar should be pretty small by default, but perhaps if your theme has a large font size then the bar is showing up pretty big. So perhaps tweaking the font size using the built-in setting is your best bet. You can try entering a small font size like 14px and see if it looks better for you.
It’s impossible for me to know though without looking at the site in question. It’s always helpful if you can share the URL in question, because it’s also possible that some CSS on your site is causing the bar to display in a way that’s not intended.
