Here is how I ended up doing it with an update cursor:
import arcpy
table = r'C:\GIS\Forbuss\Forbuss.gdb\ForbussDataTable'
nameTable = r'C:\GIS\Forbuss\Forbuss.gdb\ForbussDataTable_Statistics'
nameList = [r[0] for r in arcpy.da.SearchCursor(nameTable,'basin_name','basin_name IS NOT NULL') ]
arcpy.management.MakeTableView(table,'tableView')
for name in nameList:
select = f"basin_name = '{name}'"
fields = ['description','basin_name']
with arcpy.da.UpdateCursor('tableView',fields,select) as cursor:
rec = 0
for row in cursor:
rec+=1
row[0] = (f'{row[1]}{str(rec).zfill(4)}')
cursor.updateRow(row)
Lines 3,4 & 5 set the stage; I converted your csv into a fgdb table, and then using list comprehension and a search cursor, created a list of basin names (where the basin name is not null). Then in Line 7 I turn the original table into a table view since I'm running this as a stand alone script. Finally at row 9 I iterate through the list of basin names and with an update cursor update the 'description' field.
That should just about do it....