Buenos Aires is a huge city so there are some neighbourhoods that are very safe and other ones that could be a little bit dangerous. Violent crime is extremely rare.
Nevertheless, it is important to remember there a number of thieves who spend their time in tourist areas on the lookout for easy prey. Always protect your valuables and keep your passport and tickets in a safe at your homestay.
The Buenos Aires barrio of La Boca has seen a number of thefts of valuables just a few blocks away from the famous street Caminito. Take special care when you are in that neighbourhood. Do not go outside of the tourist area.
Taxis are found all over Buenos Aires. Taxis in Buenos Aires are all clearly marked and have a meter. In general, it is completely safe to travel by taxi in Buenos Aires, but rumors of happenings have been known to circulate. If you are at all concerned, go to the nearest hotel and take the taxi from there. Also, you can always call a taxi service from your restaurant or a public facility. The taxis ordered in such a manner are extremely safe.
Argentina is a fairly safe country for women to travel in. Besides a few quick come-ons by the local men (that are meant to be ignored), women feel at ease exploring Buenos Aires and any other area of Argentina. It is recommended for women not to wear flashy or expensive-looking jewelry and to not walk around over- or under-dressed.
Older people are very much revered and respected by Argentine culture and there is no reason for senior-citizens not to travel to Argentina.