As FCP X is currently designed, stabilizing multicam material can be very difficult if you have lots of camera angles. The only solution I know requires the below complex work-around.
1. On multi cam storyline, select clip to stabilize, then shift+f to locate the clip in the event browser, this will highlight it.
2. Cmd+1: go to event browser window, look at time code & remember it. To enable direct time code display in skimmer: View->Show Skimmer Info.
3. On multicam storyline, right click the clip, select “Open in angle editor"
4. Once in Angle Editor, press Control+P to type in the remembered time code. Don’t enter colons, just the numbers and press enter. This will jump to the base clip in the angle editor. If many sync’d tracks, you have to figure out which clip by observation but the playhead should be in the right location.
5. Blade that section of the clip, select it, then apply stabilization.
6. In upper left of angle editor timelines, press back arrow to return to multi cam storyline.
I encounter this situation frequently when syncing a lof of material using Plural Eyes and creating a large multi-cam project from that. You only know what clips require stabilizing after editing is nearly complete, so it's not practical to stabilize the base clips before this. Yet there's no way to stabilize the clips directly in the multicam storyline.
This procedure is not required for color correction or other effects in a multicam storyline.