Find an Artist
Websites like Fiverr and Dribbble provide easy access to a wide variety of creative designers. Through sites like these, you can see examples of each artist's work (to find a style that suits your preferences), discuss your project with them, ask for a quote and ultimately engage them to prepare an illustration for you (in my opinion, Fiverr does a better job of helping you through this process).
While sites like these cater to a wide variety of creative professionals (not just illustrators), a key-word search works pretty well to quickly narrow the list to relevant artists. For example, when looking for someone to illustrate children's stories, I use search words such as "children's illustration" or "child cartoon".
Specify What You Want
So how to best ensure the artist's work reflects your inspiration?
Capture What Your Vision Looks Like
One of the surest ways to minimize uncertainly is to simply draw your picture (I say this as a particularly unskilled freehand artist). Where a picture is intended to capture the essence of some related text, I like to provide that text to the artist as well. For example, the photo and text below (on the left), is my attempt at outlining the vision I had for a Wee Willie Winkie illustration. The illustration on the right shows the artist's first attempt at capturing that vision - an outcome I was very happy with.
Specify How You Will Use the Artwork
Such specifications may vary from project to project. However, I find it helpful to have a simple style sheet (like that copied below) to share upfront with artists I am looking to engage.
Having guidelines around characters, size of artwork, file type and how I intend to use the illustrations has minimized the need for revisions and the associated extra costs. For example, the '*.png' file format lends itself to transparent backgrounds, which are helpful if you later want to overlay one illustration on another to create different scenes.
Keep in mind that artists you find on the web come from myriad backgrounds and cultures. This gives you access to a wonderfully rich body of influences, which sometimes can show up in the work you receive in unexpected ways (e.g., the nature of clothing that characters wear). Equally, I find that different artists talk about the level of detail they will include (e.g., as background behind your key subject) in different ways. Given the scope of uncertainties that may manifest themselves in the final product you receive, it can be difficult to fully appreciate an artist's vision (of your vision) at a granular level without doing an initial mini project to see how you and they will work in practice..
As the illustrations below show, even a reasonable amount of guidance leaves plenty of scope for artistic licence. My original freehand for Little Miss Muffet is on the left. Each of the other illustrations (all very well done in their own style) was generated by a different artist I found on Fiverr (relying on the same set of instructions).