There is NO impact on tourists in Athens due to the immigrant situation ... arriving at the port, those ships are far from any piers any tourists would use, and those immigrants/refugees are taken to a center outside the city. Furthermore, for many decades Greece has had the record of having one of the lowest crime rates in Europe. Your only concern should be the immense crowd of other tourists you'll have to cope with in Santorini & Mykonos, the 2 most-visited islands in THE most visited time of year.
And why are you asking about bringing cash? Who has told you to do this? There is NO need whatever to bring case, NO problem with ATMs anywhere in Greece ... never was, for tourists, even during 2015's debt crisis. ATMs abound at the Airport; just withdraw 400-500 Euros and that should last you for a few days' cash needs, and use your credit cards for Hotel, car rental etc. There are only a few common-sense things to know about money/cc's & other valuables abroad (NOT just in Greece, everyplace):
• lnform your bank(s) AND your credit-card(s) issuers that you'll be withdrawing money and using CC's abroad, otherwise on your first usage outside the country they'll freeze the accounts thinking there is fraud. Tell them exact dates you'll be away.
• PLease know, when you check your monthly records upon return, any fees for usage are NOT from a Greek bank -- they are charged by YOUR home bank; the biggie banks (Chase etc) charge more, Credit Unions & online accounts charge almost nothing.
• Both of you should have your own CCs & ATM cards, just as back-up. It's also just a good idea to use ATMS that are attached to a bank, either just outside the door or in a lobby. Small ATMs in a grocery or bar may carry a charge (as they do in US as u know).
• It's not necessary to carry around everything -- passport, wallet all your cash etc -- while out & about. Leave all that stuff locked in your suitcase in the hotel, just take what you'll need for the day in a "change purse". Greece does NOT require u to have passport on hand, just I.D., and in my 12 trips to Greece have never been asked for it. On the Metro, male tourists should not carry their billfold in pants pockets (on the tourist-heavy Metro lines from Airport/seaport, there are some pickpockets, 99% of them Albanians) but otherwise, just ordinary prudence.
• You'll be taking a taxi on arrival to your hotel, and again back to airport or pier. Where is the "danger"?? I myself take the express bus from /to the airport (in 12 trips to Greece) and indeed have walked at 5 am from hotel to Syntagma Square bus stop. I lived in Manhattan for 35 years and have been in most US large city plus Europe capitals ... and I've never felt safer than in Athens.
• The fear you manifest is puzzling, makes me wonder what news sources you've gotten this bad info from -- All I can think is that you are reading some stale articles from 2015, from protests during the Euro crisis, but that is a thing of the past.