Buenos Aires is a huge city and has all the characteristics of a European capital. Places like Puerto Madero, for instance, are completely safe and you can relax and walk around, shop, dine, without having to watch out, because it's very well watched and controlled by security guards. If you adventure into some of the slum areas in Greater Buenos Aires (Ciudad Oculta, for instance), you'll certainly be at risk.
In general, be careful. Try not to "look" like a tourist much, be alert if the area you're in is not guarded, avoid places that are too crowded, like the pedestrian streets downtown (Florida and Lavalle), or if you choose to go, be aware of where you money or your purse is and watch it. Don't leave your purse on the floor at restaurants, leave it somewhere where you can feel it or see it. In short, take all the precautions you would take in any capital city in the world. You will certainly have a wonderful time. Argentina has the most beautiful landscapes I've ever seen and all kinds of climate. If you like it warm you can go to the north, mountains in the west, jungle in the east. If you like it cold, go to the south, high peaks and snow in the west, the sea, penguins and whales in the east. If you don't like extreme weather, stay in the plains, visit Patagonia, Merlo in San Luis Province, the Moon Valley in San Juan Province (it really looks like being on the Moon!) If you like the beach, go in their summer (December to March) and enjoy their beaches on the Atlantic coast.
If you don't want to travel around much, just stay in Buenos Aires. It's full of glamour and it has plenty of cultural activities. As for the budget, you'll find it amazingly cheap, considering that the exchange rate is so favourable for us now ($1 = 3 pesos) and each dollar goes a long way. And their food is delicious, especially if you like beef! When in Buenos Aires, don't forget to visit the Colon Theatre and to go to San Telmo for a tango evening! You'll love Argentina, I promise.