Is the US responsible for any crime committed by members of ships that fly the star-spangled banner?

To some extent, yes. US law applies on US-flagged ships.

https://en.wikipedia.org/wiki/Flag_of_convenience

> A ship's flag state exercises regulatory control over the vessel and is required to inspect it regularly, certify the ship's equipment and crew, and issue safety and pollution prevention documents.

Because US law is strong in this regard, the US military is by far the largest contributor to the count. Less than 200 civilian vessels are flagged in the US; https://www.statista.com/statistics/652126/us-flag-oceangoin....