You can use Export Map under the Share tab.
Set the Map display to the extent of the basemap you are interested in.The output will take on the Spatial Reference of the Map.
Most export formats have the option to:
It shows the extent of the Map that will be exported. You can zoom in/out to set the area of interest. The output is not likely to look exactly like the input and it will be fixed at the current resolution. Be sure to check:
In a pinch, you can export to PDF first, then use GP tool PDF to TIFF to render to TIFF. I found that exporting a chunk of the Imagery basemap, the png format was best but you need to check on: Write world file.
Hope this gets you started.