I read a sci fi story once where some 17th century English sailors encounter aliens and teach them Latin so they can communicate with them, but still speak to each other privately. How common was the ability to speak Latin in 17th century England?

In the 17th century it was part of normal curriculum for an upper class person to take Latin and sometimes Greek. Many books of the period were published in Latin, and it was considered the universal language of the intellectual elite of Europe. For example, Newton published in Latin.

But it wasn't commonly spoken, it was a literary heritage. The old classics and books expecting to have international appeal were in Latin. A book that was expected to only be sold locally would normally be in the native language of the author.

So the idea of common sailors knowing Latin doesn't sound right to me.