For vertically stacked open file tabs, put this in your `./atom/styles.less`

atom-vertical-file-tabs.less

atom-workspace-axis.vertical atom-pane {
  flex-direction: row;
  .tab-bar:not(:empty) {
    box-shadow: inset -1px 0 0 #181a1f;
    resize: horizontal;
    height: auto;
    display: block;
    padding-right: 1px;
    padding-bottom: 3em;
    min-width: 14em;
    &::after {
      content: '';
      position: absolute;
      bottom: 1px;
      right: 1px;
      width: 0;
      height: 0;
      border: .7em solid transparent;
      border-color: transparent @ui-site-color-2 @ui-site-color-2 transparent;
      box-shadow: 1px 1px 0 0 #181a1f;
    }
  }
  .tab-bar .tab {
    display: block;
    border: none;
  }
  .tab-bar .tab,
  .tab-bar .tab.active {
    flex: none;
    width: auto;
    max-width: none;
    box-shadow: inset 0 -1px 0 0 #181a1f;
  }
  .tab-bar .tab .title {
    text-align: left;
  }
}

Resizing the tab bar width is possible with the resize: horizontal; style. The file tabs container can be resized by dragging the little blue triangle in the bottom right corner of the tab bar (like a textarea element).

This is a little bit awkward, but maybe there's a way to run a launch script that could add/control a resize handle like normal column resizing interactions.

Update: added :not(:empty) to the .tab-bar rule so the tab bar won't invoke the ::after resize cue when there are no open files — this lets the tab-bar respect the display: none from the base tab-bar component styles for an empty tab-bar.