You can make a mask in a data frame. Go to data frame properties, specify a mask, specify the layers you want masked. In the data frame properties select the Data Frame tab, under Clip Options select "clip to shape", then click the button to specify the layer you want to serve as the mask.
If you want to specifically block out part of the basemap then you may have to erase part of a mask layer. For instance if you want a basemap that covers the US but not Kansas, you can take a map of the US, select Kansas, make a new layer from Kansas, then use Analysis > Overlay > Erase to remove Kansas from the US layer. Use the resulting layer for the "clip to shape" option. It will block the basemap and whatever other layers you want from showing Kansas.
