In Omonia, in addition to people going about their legitimate concerns, you'll find lost tourists, cripples, junkies, beggars, and displaced persons from the Balkans and elsewhere like African street vendors. You see lots of Africans selling things in Athens particularly bootleg cd's and videos.
Omonia seems to have a sleazy reputation but other than being so busy I can't really see any reason not to go there. It is interesting and has lots of stores selling inexpensive clothing, shoes, and just about anything else you can think of.
Be aware of pick pockets and you'll be fine. Beneath the square is the nexus of Athens electric train system so its easy enough to get to or depart from. I don't mean to put you off the square, but be street smart, especially at night. Don't flash a wad of cash, your $600 NIKON and use your common sense and you should be fine.
Athens is one of the safest cities I know but now as a full EU partner anyone from the EU can come down here to live and bring their nasty habits with them. 10 new countries just joined so who knows what that will mean. Anyway, in the day time, Omonia Sq. seems safe enough but caution should be exercised at night. As in any big city hustlers seem to congregate downtown.
Omonia Sq. does offer a few things worth mentioning and one of these is the 100 Aeolou Street Post office which has extended hours. Another is the internet cafe nearby on Gladstonos Street. A third is the several Bureau de Change and Western Union type places to send (or receive) money. There is a good place to buy coffee in all its manifestations called Loumidies and several restaurants and cafes as well. There is also a (new in 2006) very decent English language bookstore right in the heart of Omonia Sq. They have other languages too. There is also a Starbucks if you want to pay extra.
Omonia is also one of the two most important Metro Stations and from it you can reach Piraeus but for the airport you have to go one stop to Syntagma and then change lines to get to the airport (ATH).