When we think of Gothic literature one of the first things that comes to mind is setting. Eerie towers at night, spooky graveyards, haunted houses, deserted mansions and so on. Then we have to inject an element of the supernatural - strange sounds, unexplainable happenings, supposed ghosts. If you look at novels that are considered "Gothic" such as Jane Eyre or Wuthering Heights, these are all key ingredients.
So, when we think about "The Masque of the Red Death," we need to investigate if we can detect Gothic elements in the setting and if there are elements of the supernatural. Certainly, the setting seems to be particularly bizarre, and it is worth thinking about how it operates symbolically. However, the last, black room, seems particularly Gothic in its presentation:
The seventh apartment was closely shrouded in black velvet tapestries that hung all over the ceiling and down the walls, falling in heavy folds upon a carpet of the same material and hue. But in this chamber only, the colour of the windows failed to correspond with the decorations. The panes here were scarlet - a deep blood colour.
Add this to the light from the brazier that produces an effect "ghastly in the extreme," and you have a clear Gothic setting here. Note too the element of the supernatural, with the uninvited guest, whose costume makes him resemble a "stiffened corpse" and who disappears when challenged. An obvious example of Gothic supernatural.
So, in this story you can analyse the setting and also the element of the supernatural that is introduced through the uninvited guest to find examples of how this story is considered Gothic.