I haven’t ever published a game here (still working on my first WIP in fact), but I can come at this from the other side as a professional artist/illustrator. Apologies in advance for the novel!
Commissioning art is really one of those things where you get what you pay for. I saw @Eric_Moser say $100-$300 for a cover image, but I think that’s really rather low. Speaking as a professional, a lot of the art I see for Choice of Games/Hosted Games products seems to be on the amateur side of the spectrum, and the prices Eric quotes reflect that. For comparison, Wizards of the Coast pay $1000-1200 for a Magic the Gathering card.
Typically the process for commissioning art works as follows: you send me a message with a brief idea of what you want, how big it needs to be (whether in inches/feet for a physical piece of art or in pixels for purely digital), and how quickly you need it. Based on that, I can give a rough price estimate. From there, we can either move ahead or negotiate ways to bring the price down (typically more freedom for me to do what I want, a longer time to do it, or a more simplistic concept). My price usually includes 3 minor revisions with the option to pay for more if you feel they’re needed. The way the process is set up though, there’s many opportunities to catch issues early on.
Moving ahead, I’ll do 3-5 thumbnail images of my ideas for the project based on your description. These are small (usually smaller than an index card) and offer distinct variations on your theme. You choose which one you like best and suggest changes- this is the time for major revisions (ones that don’t cost you extra!), as no major work has been completed yet. Things like “the knights need to be in a specific type of armor” or “this character is actually taller than this character and is black, not white” or “the architecture should be Greco-Roman, rather than Medieval” are appropriate at this stage. This is also where I ask for a 50% deposit up front, non-refundable (this serves to protect me in case you’re forced to cancel the commission after I’ve already done a considerable amount of work). The contract is signed at this point as well.
I next do a detailed version of the drawing. This is pretty much how the piece will look finished, albeit it lacking color and rendering. This is when things really get locked in, so it’s important to communicate any issues before painting actually begins.
Once the drawing is approved, I set to work on the final piece. Depending on how long I think it will take, I may check in once or twice with in-process shots. If it’s simple and relatively fast, I will proceed to the finish. I’ll send you a low resolution image of the final piece (not pixelated but not high resolution) and give you a chance to approve the piece. This is where the free three minor revisions come in. Things like “could you tone down the red in this character’s hair” or “the sky should be a bit brighter” or “could you make the armor be a bit shinier.” With it approved, I ask for the remaining 50% and then send you the high resolution file (or ship you the artwork if it was a physical piece). At that point, everything’s done! 
Also of note, an artist doesn’t necessarily have to sign all the rights to the image away- that’s another way to bring down the cost of the project. For instance, I could give you the rights to the image for gaming purposes, but reserve the right to produce prints and sell merchandise with the image on it. Limiting the time you have rights to the image is also another avenue. You might contractually only get full rights to the image for two years, and then they revert back to me, for example.
I agree completely with Eric: games with low quality art tend to push me away. Typically it’s a really good response here or in the reviews that cause me to go back and take a look.
An important thing to consider is that an illustrator is not automatically a graphic designer. When an artist is hired to do a book cover, for example, they don’t do the text and jacket design, only the illustration. The graphic design is handled by a professional, usually in house, who specializes in that field. So if you’re hiring someone to do art, don’t assume that they’ll do the text/logo as well! They might be a skilled graphic designer, but in my experience most illustrators are not.
If you made it through this, more power to you, and I hope it was beneficial! For those curious, my art can be found here: matthewjhuntley.com Shameless plug, but I’d love to work with someone on a game cover if the project is right! Feel free to reach out!
I find that my clients are always pleasantly surprised by the sketches I present them, and I almost always hear “that’s better than what I pictured!”