This website is sponsored by the British National Maritime Museum and features 39,000 crew lists containing over 750,000 names dating from 1915. The database is searchable by sailor's name, rank, birth place, and vessel name. There is a separate search form for vessels only.
The home page offers a search form asking for surname only, but scroll down a bit and you will see the more detailed form.
The hit list will be cards giving sailor and vessel information. Click on the card to see the digitized crew list. The crew lists can extend for multiple pages.