Answering the question in the title until one wants to shout: huge! This is one of those elements that create the character of a story and its atmosphere. The best script, but made with a technique that does not reflect the atmosphere of the story, simply will not defend itself.
The style of an illustrator consists of a myriad of elements, from the way the lines are drawn, through colors - their use or not, shapes, proportions of the characters, realism, or, on the contrary, surrealism and related proportions. All this, as well as the technique used: paints, crayons, markers, pen, pencil, thin pencil, or collage, has an overwhelming influence on the style in which the final comic will be.
The topic solves itself, so to speak, if the comic writer is also the author of the drawings. On the other hand, if the comic is created by a duo, it is the responsibility of the screenwriter to choose the right artistic partner for the story.
There is in the work of cartoonists that distinctive feature, thanks to which a single glance is enough for a skilled eye to recognize the author of the drawings. It is good for a comic book writer to be able to turn this recognition into an advantage - unfortunately, it happens that a particular style contributes to the pigeonholing of artists. Therefore, it is worthwhile, when establishing cooperation with a particular artist, to write a story to his needs as well, perhaps waiting for a chance to showcase his talent in something other than, for example, exclusively associated with his dark style of crime noir. Art likes to surprise, so it's worth thinking carefully about how to capitalize on a cartoonist's recognition.
Illustration, especially comic book illustration, and therefore, as a rule, pop culture, gives great room for experimentation. Therefore, instead of locking yourself into a style you've invented, it's a good idea to discuss the matter with the illustrator. It is possible that he will see the subject slightly differently, or just the whole story will inspire him to experiment with his style. Writers, do not be afraid of these experiments - it is from them that the best works can come out, full of artistic freedom and unprecedented solutions.
Any written work (or written and drawn, like a comic book or graphic novel), to open the door to the world of imagination, must have its atmosphere. It's what makes some books far from being literary works, and yet they become reader bestsellers. In the case of comics, the comic writer has as many as two powerful tools at his disposal - one is the plot and narrative, and the other is the very image.
In today's culture, in which the image acts as a full-fledged carrier of content, it is good to lean on this element and pay more attention to it. From the perspective of a comic scriptwriter, it is worthwhile to choose the collaboration on a particular story in such a way that the final style of the artist pins with what is to be conveyed. And there are no better or worse styles - each simply represents a different form of artistic expression, and the scriptwriter's head in this is to properly assess the mood of his story and, with the help of the illustrator, create the right atmosphere for it.
In conclusion, the choice of style of a comic book illustrator is more than simply how to draw characters, scenes or take perspective. It is a powerful artistic tool that influences the reception of a comic book, builds the identity of the creators, and is an important part of the narrative.