Yes, it is possible to determine the date of an image (specifically, any region of the satellite imagery) in the Imagery basemap. The World Imagery basemap is a compilation of imagery sources for the world and it is comprised of datasets that cover both large and small areas.
Note: The basemap is made up of several imagery tiles. Different tiles and different zoom levels were taken on different dates. Updates occur on the different images within the basemap, and there is no actual known cycle for this activity. Refer to FAQ: How frequently is the World Imagery basemap updated? for more information.
In ArcMap, the Identify tool can be used to determine the date the satellite image was taken in the World Imagery basemap.
To view the date, click the desired area using the Identify tool. The date is displayed in the Identify window as seen in the image below.
To use this tool, refer to ArcMap: Identifying features.
Note: For viewing the date of an image from the World Imagery basemap as well as its metadata in ArcGIS Pro, refer to How To: View the World Imagery basemap metadata in ArcGIS Pro.