There is nothing to toggle on or off in the table of contents... except for the whole basemap... at least with the world and streetmap files... Have you looked at ArcGIS Online data sources?
Searching in ArcGIS online has always been a challenging task for me - a lot of noises. It would be great to turn off or change the color of the street labels in Vector basemap, programmatically or otherwise.
I ended up creating my own basemap (second pic) because it looked too rough to have my street labels on top of Esri's (first pic).