I'm far from being a naval expert, but two things come to my mind instantly:

Sailing ships are not that easy to stop or to turn and manouvre to a specific spot, so you would need a lot of time and most probably smaller rescue boats brought to water, which would take even more time. So it's most probable not possible during an ongoing battle. And after the battle is over, the ships will have moved on, as battles were not fought stationary but while moving. And then, even finding the spot where another ship sank would be very difficult.

Secondly, most sailors couldn't swim, so unless they found some planks to hold onto, they would simply drown.

Just my thoughts, no sources to back this, so maybe I'm completly wrong. But I strongly doubt it was possible or common to save sailors after battle.