From Kusadasi you can take a 'dolmus' to Ephesus. The dolmus are minibuses that pick up people along the regular bus routes till the minibus is full. They run very frequently and have no fixed stopping points, so you can take them everywhere and you can also get off wherever and whenever you want.
For Ephesus take the dolmus with final destination Selçuk. The destination is well marked on the windshield of the minibuses. You can take a dolmus close to the port, but there's a chance that the bus will be already quite full. In the worst case you have no seat. It's therefore better to go to the starting point of the dolmus. This is about 1 kilometer from the cruise port at the roundabout of Adnan Menderes Cad.
The dolmus doesn't bring you to the entrance of Ephesus, but to the taxi stop on the main road about 1.5 kilomterer from the archaeological site. You can walk there in 10-15 minutes. For a taxi you pay around tl15 per taxi (4 persons, one way). The bus ride from Kusadasi to Selcuk takes about 30 minutes and you pay TL11 per person (one way). You pay cash on the bus.