Make sure you have appropriate travel insurance before travelling to Argentina - it's a good idea to get insurance which covers medical expenses abroad and if you plan on doing any extreme sports ensure that you are covered. Be sure to look at various travel websites for health care advice (including required injections and precautions) including those produced by the government of your country. The World Health Organisation provides extensive information on epidemics and diseases throughout the world: http://www.who.int/ith/en/
Perhaps most problematic in Argentina is Dengue Fever which is a viral infection transmitted by mosquitoes. There is currently no vaccine for Dengue but make sure you pack plenty of insect repellents and associated protective measures to avoid being bitten.
Seek advice from locals on the quality of water and whether it is suitable for drinking. Tap water in the richer areas of Buenos Aires is usually fine to drink but this can vary. If you're staying in hostels ask the people in reception for advice and if you're in doubt purchase mineral water which is fairly cheap. Things are a little different when travelling in more remote parts of Argentina where the water is generally not safe to drink - it's a good idea to always drink mineral water in these locations.
The sun tends to be very strong during the summer months so it's important to take the usual precautions: wearing a hat and using copious amounts of sunscreen and lip protection. Try to avoid spending time in the direct sun, especially from 1100-1400 and take lots of fluid to avoid dehydration.
The best way to avoid stomach upsets is to drink bottled water (especially in the interior) and to avoid purchasing foodstuffs (particularly uncooked fruit and vegetables) on the street. However, when travelling it is quite possible that you'll get some form of stomach upset. If you are affected get plenty of rest and drink lots of liquids. Oral Rehydration Solutions (ORS) are available from pharmacists and can help you to replace the fluids lost. If you are affected for more than three days and/or you have a fever, you should visit a doctor. Pharmacies are generally very well stocked and easy to find particularly in larger urban areas of Argentina.
Those staying in Buenos Aires and nearby won't need to worry about altitude sickness but if you do decide to visit places in the west or northwest of the country (i.e. in or close to the Andes) you need to be aware of the associated dangers. Some people experience discomfort at higher altitude - the symptoms include headaches, dizziness, stomach upset and tiredness. These symptoms can be managed by reducing alcohol intake; drinking lots of fluids; eating light meals and getting plenty of rest.
There are no extraordinary dangers which you need to worry about when travelling in Argentina. However, this is a society with huge social inequalities and as a result incidences of petty crime are not unusual, especially in the streets of the larger cities. It's necessary to be vigilant when walking around crowded areas - keep your belongings securely packed away and avoid walking alone late at night (although you will find lots of people on the streets into the early morning which provides a certain feeling of security). Perhaps the greatest danger that you will encounter is the traffic, particularly on the busy streets of Buenos Aires. Always be alert as the drivers are not known for their courteous manner towards pedestrians!