After completing all other achievements, I only had about 40 left to recruit. I then loaded up Sequence 3 Memory 2 "Now Hiring" (the same one used for the Siren Song achievement) and just repeated this until I had enough.
I got approx. 20 new recruits doing this mission - the one big downside being an unskippable cut scene at the end of the memory.
While I am not certain if this is quicker than taking down Man o Wars or not, by this point of the game I was sick of boarding ships, so I preferred this method.
Updated thanks to STAMMxTHExMAN: To check your crew member progress, check the Statistics section of the Database.
Updated thanks to mike cupcake: "You don't need to watch any cutscenes, just restart the memory once you've done the first batch of rescues."