Mr Drax visited the volunteer-led attraction as part of English Tourism Week and to learn more about its £14 million annual contribution to the Purbeck and Dorset economy.
The award-winning five and a half mile heritage line runs from Norden Park and Ride to Corfe Castle, Harman’s Cross, Herston and Swanage.
The train now carries more than 210,000 passengers a year after being rebuilt from nothing in 1976.
Mr Drax arrived by steam train at Swanage station and was welcomed by the Mayor of Swanage, Ali Patrick, as well as tourism staff from Swanage and Purbeck councils.
He said: “The Swanage Railway plays a vital and integral part in promoting tourism in Purbeck.
“Those romantic days of steam are played out every day on this scenic stretch of line which is loved and maintained by a band of railway enthusiasts who have Purbeck’s interests at heart.
"This team has worked tirelessly to provide a service for residents and tourists alike and my visit to the Swanage Railway was a highly enjoyable, informative and action-packed afternoon.
"The Swanage Railway is right at the heart of life in Purbeck and offers an unforgettable experience to tourists who come to Dorset to savour our rural culture.”
Mr Drax spoke to some of the volunteers that run the train services and saw behind the scenes of the Swanage Railway.
He was hosted by Swanage Railway Trust Chairman Gavin Johns and Swanage Railway Company Chairman Peter Sills.
Mr Johns said: "Everyone was delighted to welcome Richard to the Swanage Railway to see the results of all our hard work over almost 40 years and learn about our plans for the future which will continue to benefit the Purbeck and Dorset economy.
"The Swanage Railway Trust is fulfilling its long-term objectives of funding and creating an educational living history heritage railway from the 1950s that is fit for the new Millennium with our Project Wareham and plans for new train maintenance facilities."
Meanwhile, Mr Sills said: "The Swanage Railway enhances the experience of visitors to Purbeck and encourages them to extend their stay in the area.
"The Swanage Railway also helps bring more business to local hotels, restaurants, bed and breakfast establishments and campsites."
English Tourism Week runs from Saturday March 29 to Sunday April 6.