If you want to label some of the objects in a base map, but not all, you will need to use the attribute table rather than the built in label properties. Using the attribute table, you can create a new attribute for that object and set the label source to be that attribute.

The steps to do this are below:

1. Select the Base(vector) layer in the Contents window and click Map Tools | Layer Tools | Open Attribute Table.
2. In the Attribute Table, click the Add Field button ().
3. Give the attribute a name (e.g. "Label") and click OK. The attribute field is added.
4. In the Plot window, select the first object you want to label to highlight it in the Attribute Table. Simply enter the attribute value you want for that object in the new field.
5. Repeat step 4 for each object you want to label, selecting the object and entering in the label in the Attribute Table.
5. If desired, click Save to save the symbology and attributes you created. Apply to future projects with matching attributes using the Load button. 
6. When finished, you can close the Attribute Table by clicking the X if you wish.
7. Select the Base(vector) layer, and click the Labels tab in the Properties window.
8. Set the Label source to be the attribute just created.

Updated November 10, 2021