In addition to the excellent points made about the Gothic tone that is reinforced by forbidding weather of so many scenes, the storms and tumultuous weather serves as a reminder of the night on which the creature of Victor Frankenstein came into being. Much like a motif in music, the threatening weather of that fateful night is repeated to connote the sinfulness of Victor in "playing God" and fashioning a creature of his own.
That it is cold and so barren in many scenes is also significant. Rather than the typical forests and dark castles with their subterranean passageways and rooms of the Gothic genre, Shelley's novel has the brutal cold and iciness of the northern climate suggesting he evil. This employment of the lack of color and the lack of warmth is analagous to that of Herman Melville's great metaphysical novel, Moby Dick, in which the absence of color is evil. Thus, the snow-capped Mount Blanc are reflective of the absence of human warmth, the absence of any sentiment, any humaneness with which the creature is met.
The instances of bad weather serve as allegorical literary techniques that attribute foreshadowing, atmosphere, and Gothic elements. The foreshadowing comes as a result of the internal turmoil of the two main characters, Victor and the monster. When the action becomes climatic, the use of nature as a punisher is part of the Romantic characteristic of Gothic literature which intends that men and nature are fused together, and what man does, nature will re tribute. The atmosphere is achieved by presenting bad weather as an inconvenient, limiting, exasperating, annoying, cold, wet, despicable, reductionist, and untimely event that goes hand in hand with mental limitations, exasperation, annoyance, coldness, and hatred.
Finally, the Gothic element of darkness, shock, inevitability, nature against man, and man against himself are enhanced with the use of rain and storms as things that push the psyche of the story forwards towards nostalgia and pessimism.