The lien on ownership and the responsibility for the taxes are established on January 1 for the following tax year. The bill for the coming tax year is issued to the owner of record at that time, and that individual is liable for the taxes even if the boat was sold soon after that date.
When the boat was sold just before the lien date (December 30, for example), the new owner would be liable for the new bill. If you were not the owner on January 1, but the bill was issued in your name, then you need to contact the Assessor's Marine Division to resolve the ownership issue at (415) 473-7208 (9 a.m. – 4 p.m.).
The Assessor does not prorate the assessments.