Actually, Kelly - it depends! Eric, is the apartment in Greece a second home or are you actually moving?
What ISP do you use in Basel? Do you have broadband, DSL or dial-up? I don't know anything about the European ISPs, and therefore don't know if there are providers that operate in more than one country. With dial-up, here in North America, there are some ISPs that have local access numbers all over the US and Canada. My in-laws, for instance, have hotdial.net, which has over 10,000 local access numbers. (AOL is another service with widespread local access numbers) They can use their iBook at home and when travelling. I've also used their account when travelling, with their knowledge, of course, because my ISP is broadband only. Some broadband services apparently also have dial-up numbers that customers can use when travelling.
So, if this IS a second home, you need to find out what your current ISP offers. If they have no options like remote dial-up, then you will need to get separate internet service in Argos. If you're moving, then as Kelly noted, you'll definitely need a new ISP in Argos.
Generally, your friends who are travelling are using the hotel's internet service to log into their own ISP's web mail or their own company's email etc. You can do that too from Argos, but you do need an actual internet connection. If there is nothing like a local dial-up number for your current ISP, then you'll have to decide how much you want to spend and what sort of service you want at the new apartment.