Hi
I am Greek and I have grown up in Athens (20+ years). I just finished my studies in Pireus (10km from Athens Center), however the last year i leave abroad.
I can say that Athens is very nice city but quite crowded. It's fool of nice pubs and bars and the student life is quite nice.
Athens is quite multicultural and many people from US live there (many tourists come as well). Most of the students are really helpful with the non-native students and defenatily Greeks speak better English than Italians :-p (i have an Italian friend and colleague next to me already seen the post).
Furthermore, winter does not really exist and only 15 km from city centre you can find amazing green-blue colored beaches.
Regarding your budget, 1500usd is more than enough. Actually you will find an apartment with ~300euros in very good area cose to your university (i guess ΕΚΠΑ or ΕΜΠ). I would reccomend areas such as Pagrati or Ampelokipi (many americans live there). Also the public transport is really good and very cheap (only 15euros for monthly student card) for everything to everywhere inn Athens (and the metro is brand new comparing to other i.e London where a single ticket souldn't cost more than 0.4 euros, but it costs 4 euros.....).
About safety i do not know what you have read but it quite safe (no one never attacked me or stole my phone or smth similar) however it's massive city and you have be carefull with your belongings and avoid going to specific areas (ghettos) especially in the night.
Regarding, crysis since you are student and you do not want to live your entire life you should not be cocerned about it. It is true that people are not as wealthy where in 2009, that many are struggling and that the unemployment is pretty high but do not believe the media (thankfully they have stopped talking about us the last months).
The education it's absolutelly free (at least for Greeks), and the level is pretty high and quite demanding. I did my BSc in Athens but i chose to do a Masters abroad. I am more that satisfied from the academic level of Greek institutions.
I miss Athens but i am going for vacation in August
)
This is it
Good luck, regardless your choice