Your quote characters are wrong as shown when I print out the string:
explorer "D:\SBC\01-03IRE\15-Daily_Count\"+current_date+"\"
Change the double quotes around
current_date
to single, thus:
current_date = datetime.datetime.today().strftime('%d-%B-%Y')
subprocess.Popen('explorer "D:\\SBC\\01-03IRE\\15-Daily_Count\\' +current_date+ '\\"')
^ ^