Flying to Bari is definitely the best option if you're trying to reach Dubrovnik or the surrounding area.

Bari - Dubrovnik by Ferry
Jadrolinija operates an overnight ferry on this route once a week during off-season, and several times a week during summer.

C/F Azzurra Line The Italian passenger shipping line which maintains for the 5th year the international ferry line Bari - Dubrovnik, June to September 2006.

Flights from the UK to Bari
Ryanair have flights from London Stansted

Useful Travel Links
Bari Airport Includes information on getting to/from the airport
Trenitalia Train timetables for journeys within Italy