This page will explain how to add a tile layer as a basemap to your web map on ArcGIS Online (AGOL)
Navigate to your desired Tile Layer in AGOL
Note: It should be located in the User Group
Open your Tile Layer in the Map Viewer OR Add the Tile Layer in your existing Map Viewer
Select More Options for the Tile Layer
Select Move to Basemap
The result of moving the Tile Layer to a Basemap will look like this
If you have two Basemaps, please remove the undesired Basemap by selecting More Options and Remove
Save the Web Map