I am trying to create a map using the ESRI Basemap on ArcGIS 10.3. The Problem I am having is that the area I want to map is covered by cloud, making details impossible to see. Is there a way to change the date on the ESRI Basemap, to a previous date so that I can eliminate the cloud cover?
This is not supported out-of-the-box in ArcGIS, but there is a commercial option called Nearmap which allows this - see https://go.nearmap.com/esri
To be able to do that you would need that particular basemap service to support historical imagery, and for your client to be able to control the dates of imagery provided. You could ask for this in an ArcGIS Idea.
To see an example of an imagery service that does for a local geographic area you could look at Aerial Comparison through Time - Historic Imagery whch is a:
Map showing Cottonwood Heights over time using aerial imagery