On both the Disney Wonder and Disney Magic there are Windows based computers located in the Promenade Lounge (deck 3, mid-ship on both the Disney Magic and Wonder). The Promenade is open 24 hours a day.
There is also the Cove Cafe on deck 9 for adults that offers internet access and teens can access it on Aloft (Disney Wonder) and The Stack (Disney Magic).
The access speed on both the Disney Magic and Disney Wonder is acceptable. It is not as fast as cable broadband speed, but much better than dial up. Remembering that you are aboard a ship, often at sea, means that the internet service is often pretty good.