"The Devil and Tom Walker" has elements of both Gothic and Romantic literature, partly because the influences of these genres overlapped the time period in which it was written. Romantic literature is usually more concerned with nature and the role of man in it, while Gothic literature has more to do with reflection upon man's mortality, horror and the supernatural, though it sometimes borrowed Romantic elements. "The Devil and Tom Walker" can be considered an example of both styles, depending upon which elements are prioritized.
The most prominent Gothic setting is the ruined Indian fort; Gothic literature often involves ruins or other decrepit monuments for a variety of reasons, such as their symbolic representation of the mortality of human works, and their eventual overthrow by nature. Another element is the location of the fort within the swamp; being isolated and in an "evil", foreboding place, the fort represents a bastion of wickedness that is separate from the "healthy" part of the world, which draws a comparison between the physical isolation of the place and the metaphorical isolation one experiences by going beyond the boundaries of good conduct.
Another Gothic element is Tom's first house, which takes the place normally occupied in a Gothic story by something like a ruined castle. Tom's house is an emblem of sterility, and offers no cheer or comfort, nor do they have any friends or welcome any travelers. This, like the swamp, symbolizes how Tom and his wife are cut off from the world, and their isolation, rather than evoking a traditional Gothic response like horror, is turned more to pity and disgust because they aren't very likable characters and we would probably prefer someone like them to be removed from us as well. This also points to one of the factors that makes this story something less than a Gothic novel; it lacks a good character upon whom the isolation, horror and morality themes can fully reflect.