How to change the gutter background colour in VS Code?

You can change the gutter's background color (or colour) in settings.json. This was added in May.

"workbench.colorCustomizations": {
    "editorGutter.background": "#000000" // your color here
}

or you can add

"editor.rulers": [ 0 ]

This won't change the background but it will add a line between the gutter and the editor.