Someone set fire to the canopy at the front of the boat in Aspley Marina, Huddersfield, on May 21.
A woman and man were sleeping on the boat at the time and were awoken by the fire, which they managed to put out.
While outside they spoke to a man next to the boat who told them his name was Francis and who said he was going to go search for suspects but then left the scene.
Det Con Lisa Saberton, of Huddersfield CID, said: “We are investigating this offence and are issuing an E-fit of a man we wish to speak with in connection what took place.
“The male, who was present at the scene, told the occupiers of the boat his name was Francis, and said he would search the area but disappeared.”
“He was described as white, in his late 30s, and medium build with short dark hair and appeared to have a slight accent.
“We would ask this man to come forwards or for anyone who may know his identity to contact us as we wish to speak with him in connection with what took place.”
She added: “This was clearly a very serious offence which certainly endangered the lives of the two persons aboard and caused serious damage to the boat.
“Anyone who has information is asked to contact Kirklees CID on 101 referencing crime number 13180243275. Information can also be given anonymously to the independent Crimestoppers charity on 0800 555 111.”