The Thales VesseLink modem they used consumes 65W nominal/120W maximum. It offers a connection speed of a couple hundred kbps, so sending up a video file of a fixed size will require it to be on for quite a while - Assuming 200 kbps average, and a 360 MB video, that's 4 hours of uploading or 260 Watt-hours. Also, it's 12x9x2", and weighs 7.5 lbs; this is a boat not a hobby quadcopter. 260 Watt-hours is a lot; that's like 3 laptop batteries, but that's still smaller than the modem itself.
Starlink does consume 100W, but offers a connection speed of about 200 Mbps. The 360 MB video upload could complete in 14.4 seconds, which consumes 100 W * 14.4 seconds / 3600 seconds/hour = 0.4 Watt-hours. It is significantly larger, and it would probably have a harder time handling rough seas (not to mention saltwater intrusion), but that's a lot less power.
Whichever modem you're using, you'd want to turn it on infrequently.
Edit: The Saildrone product brief is here:
https://assets.website-files.com/5beaf972d32c0c1ce1fa1863/61...
It describes a 23' or 7m boat. The 33'/10m larger version has 300W continuous sensor power/2kW peak available from the solar panels, which appear to be of a comparable size to those on the Saildrone.
It will be a life-changing event for maritime robotics, assuming they don't get too greedy.