Chief Kankasem Meesuk said an initial investigation into the clip posted by Facebook user Sea Man found that it was a long-tailed boat but he could not clearly see the name.
He could not say whether the incident took place off Koh Ngai in Krabi, he said.
The chief said that around the island, the national park has prepared buoys for boats to tie to prevent damage to the coral reefs.
The clip showed tourists on board as the anchor was dropped. The video also shows the anchor underwater being dragging back and forth along the coral reef.
Several netizens called for the authorities to take action against the skipper.