Greece is coping amazingly well with the immigrant challenge (despite the shameful lack of support from the major European countries whose open invitation to migration started the surge) ... virtually all the immigrants (from the war-torn near east, and others fleeing wars in Aftrican nations) are being housed in camps up by the northern borders, on an eastern island, and -- on the mainland -- in a former airport an hour or more outside of Athens. In parts of Athens, (mainly fringes, outside of the central tourist areas) one may see some beggars, and other people selling items laid out on sidewalks -- these would be illegal Albanian immigrants (a longtime component in Greece) and some Romani (gypsies). These would be the source of pickpocketing (chiefly on the Metro from airport or seaport) and "scams" -- petty crime is not a a common thread in Greek culture (as may be the case in Southern Italy, for example).
As for general safety in Athens, all I can tell you is that I've lived most of my life in America's big cities -- New York, Chicago, and now Philadelphia, and have stayed in most of Europe's major capitals. I have never, repeat never felt as safe in those as when walking around central Athens ... early or late evening, alone or with a friend -- and in my 12 trips I've stayed in Athens at least 17 times. And as for the islands ... nothing to report! In fact, my island hoteliers tell me that any petty crime in the summers almost always can be traced to young vacationers (who don't seem to have the same respect for personal property that you & I grew up with).