The Foreign Secretary has travelled to Anguilla and the British Virgin Islands to see the destruction wrought by the storm and observe the Government's aid efforts.
Ministers have been criticised for a sluggish response, but Mr Johnson dismissed suggestions his visit was a PR stunt.
Speaking to Sky News' Foreign Affairs Editor Sam Kiley on an aid flight to the region, alongside Royal Marines and medics, Mr Johnson said: "This has been long-planned ever since the hurricane broke.
"They've never seen anything like it for 150 years.
"People don't realise that these are British people, these are British islanders and we have a duty to them."
On his arrival in Anguilla on Tuesday, which he will tour on Wednesday before travelling to the British Virgin Islands, Mr Johnson said it was "clear this place has been through an absolutely hellish experience".
Work is needed to fully restore power as well as the island's hospital, airport and schools.
The Foreign Secretary's visit was praised by Anguillan chief minister Victor Banks, who said it "sends a very positive signal to Anguillians that the British are serious about their response".
Mr Banks revealed the UK Government had "responded within hours of the passage of the storm", adding: "By midday the next day, the Royal Navy was here with a ship."