Dubrovnik to Budva
The company Forline Shipping Ltd. operated a water taxi / fast ferry from Dubrovnik to Budva in 2018. For unknown reasons the ferry was discontinued, despite the fact that this ferry line actually made it much easier to travel from Croatia to Montenegro.
As for now the cheapest way of travelling is to take the bus timetable you can find here. As an alternative you can book a private transfer here.
From Dubrovnik airport, you either have to go Dubrovnik to catch the bus or book a transfer from the airport to Budva or other locations in Montenegro.