This may be a generational thing, but the original classic horror movies were in b&w and were inspired by German Expressionism movies and art (and much of the art were in monochromatic forms like woodcuts, ink drawings, etc.) Darkness and shadows are the foundation of the horror genre. Color can be expressive but often it also is "pretty" and therefore a distraction, hence why modern horror in color tends to want to minimize it. It's tricky because pastel colors are not the same thing as a b&w image in terms of dramatic power, so some filmmakers will compensate by emphasizing cyans, let's say, rather than have a pallet of pastels running around.
You saw in the 1990's the use of the skip bleach process, and other processes that left black silver in the image, as a way of creating that texture of b&w photography while also desaturating the image -- "Se7en" is a good example. "Sleepy Hollow" is another. Both used the process with a monochromatic production design.
In terms of production design, going for a b&w look is hardly a low-budget choice, a low-budget compromise would be to not stylize the color scheme and just use what reality throws at you.
The thing to remember with horror is that there is not one single approach to the genre. Some go for that painterly German Expressionist look and others are more realistic, like "The Shining" (though at the climax, there are some classic horror movie lighting moments). Some subvert genre expectations and some play up to them. Look at Hitchcock's works -- through most of the 1950's, he was interested in creating suspense and tension in a mundane or pleasant environment, what he called "a murder in a field of daisies" (I'm paraphrasing) but then he made "Psycho" and went for all of the Gothic horror visual elements -- besides shooting in b&w, you had the classic scary old house on the hill with storm clouds behind it, the corpse in the basement, etc. And yet within that, he also set the famous murder sequence in a brightly-lit white bathroom.
And many horror movies go from a naturalistic look to an expressionist look as part of the journey of the narrative -- the happy couple moves into the happy neighborhood, everything is sunny and colorful, and then bad things start happening and by the end, they are running around a moonlit house with lightning flashes and big shadows everywhere, etc.
It comes down to the whole notion of "what do people find scary?" and "how do I create a feeling of dread?" -- darkness and shadows are a classic tool in this case. Same goes for art direction -- what is scarier? A children's playroom in bright pastels in sunshine or some dark, damp basement that is moldy and full of cobwebs? You're not trying to create pretty pictures, you're trying to get under the skin of the viewer and unnerve them.
So there are plenty of artistic reasons for a monochromatic color design or a dark image that have nothing to do with budget.
Now in terms of shadowy, "crude" lighting, there is some backwards logic here -- if you have a tight budget, then crime and horror are good subjects to shoot because they naturally lean towards lower light levels or more shadowy lighting, there is less need to make things pretty or glamorous. So it's not so much the tight budget determining the look as it is determining the genre which allows that look.
I've found in color-correcting a horror movie that when you take a scene and decide to time it darker and with less color, it is because the question of "is it scary enough?" has come up, it's not because of some mistake in lighting that needs to be fixed.