OK - let's try a couple of other things..
1: Log out of ArcGIS Pro and then log back into it, That may refresh your connection to your data Portal.
Once you've done that click the Basemap button to see if that accesses the basemaps associated with your portal.
2: Which Portal are you signed into?
To check go to the Project Tab > Portals option. This should tell you that you are logged into an on-premise or online portal. This is where you are referencing your basemaps from.
You might need to re-assign your active portal to ArcGIS Online, for example https://www.arcgis.com
Have a chat with your Portal Administrator at your school / college / university about what your active data portal should be.
3: Can you access base maps from the Living Atlas of the World? This might be a temporary solution while you figure out step 2.
- So From ArcGIS Pro's Catalog pane click Portal tab > Living Atlas icon (far right).
- In the Search box type "topographic" and press <enter>.
Hopefully you should see the "World Topographic Map" as the 1st result.
- Drag and drop the data set into your map.
Fingers crossed that will help you out.