The most practical base is the island of Lipari. This is the largest island with the most to offer - a pretty, reasonable-sized town with a choice of accommodation and the busiest port for hydrofoil and ferry connections. You can make daytrips to the other islands from here. Panarea is the most chic island, small and exclusive with prices to match. Stromboli is for thrill-seekers wanting to climb the volcano, though apart from the hike and a boat trip to see eruptions there isn't really that much to do. Vulcano has famous sulphurous-smelling mud baths, a gently fuming volcano and beaches. Salina is green and natural and a good destination for a quieter stay. Alicudi and Filicudi are the most remote; two extinct volcanoes emerging from the sea and feeling like the end of the world; transport on Alicudi is by donkey.
In order to explore each of the islands thoroughly, you would really need to base yourself overnight on each. Ferry timetables and tourist excursions can leave you feeling rushed on day trips. Obviously, though, this wouldn't make for a very relaxing holiday, so most tourists prefer to choose one island base and pay short visits to the others.