As far as the Jersey authorities are concerned, British citizens definitely do not need a passport. See the official tourism website here:
http://jersey.com/English/transport/Pages/defa ult.aspx
All airlines require ID before you can fly with them. They all accept an EU (photocard) driving licence (or, of course, a passport). Some airlines won't accept anything else but several will accept any form of photographic ID, such as a bus pass. Use this site to identify which airlines fly to Jersey from an airport convenient to you
http://jersey.com/English/transport/byair/gett inghere/Pages/default.aspx
and then consult the website of the relevant airline(s)for definitive information.
Ferry companies do not usually require ID, and this seems to be true of Condor Ferries, who operate to Jersey from Weymouth, Poole and Portsmouth:
http://www.condorferries.co.uk/Trip/
Chris