OK, so if you are using the Classic Shop template, you can try adding this rule to your custom CSS (Appearance → Customize → Additional CSS):
body.home .fusion-header,
body.home.fusion-body .fusion-is-sticky.fusion-header-wrapper .fusion-header,
body.home .fusion-secondary-header,
body.home .fusion-secondary-menu-icon,
body.home .fusion-main-menu .sub-menu,
body.home .fusion-main-menu .sub-menu .current-menu-item > a,
body.home .fusion-main-menu .sub-menu .current_page_item > a {
background-color: rgba(255,255,255,0.5);
}
The body.home part of each selector will insure that the rule only affects the home page. The important part is the rgba color function. There are four parameters. The first three control the amount of hue in the color (red, green, and blue), and the value can range from 0 to 255. When all three values are 0, you get black, and when all three values are 255, you get white.
The last parameter is the opacity (transparency). It can range from 0 (totally transparent) to 1 (totally opaque). You can think of it as a percentage, so that a value of 0.5 means it’s 50% transparent. Right now, it’s at 0.5, so it’s 50% transparent. You can change it to 0 if you want to see how it would look totally transparent. Or change it to some other level of transparency.
Note that you will not see the transparency until you scroll the page down and you start seeing objects move under the header. That’s because the object behind the header is white, so even though the header is transparent, you won’t see the transparency until there’s something else behind it that’s not white.
You specified that you wanted the menu background to be transparent, but I wrote the selectors for the entire header; I thought it would look a little strange to just have the menu part be transparent and everything else in the header white (like the site title and tag line), but we can try it if you want.