Board game art has come a long way since the clipart looking Monopoly. Many hardcore gamers simply buy games because of the art. I don’t agree with this practice but this shows how seriously you should take your game’s artwork.
There are three types of artists you will be dealing with:
Illustrators: are probably how you think of artists in the traditional sense. They create the actual drawings. Characters. Landscapes. Whatever. They do the drawing. They create the actual art piece.
Graphic Designers: are responsible for the orientation and layout of those art pieces. They are also typically responsible for text and fonts. And are usually more equipped to handle exporting files to a manufacture’s specifications.
3D Sculptors: are able to design 3D models and miniatures. If your game has figurines, you’ll need someone to design a 3D file. This file will be used to make the molds. These artists can reference art from your game or can simply create their own designs.
Many people confuse illustrators and graphic designers. To give you an analogy of these two professions: an illustrator is like a furniture maker. They’ll make you a couch, chair, and table. While a graphic designer is like an interior designer. They’ll arrange that furniture within a room.
Some artists have both skill sets. They can illustrate and graphic design. But they’re rare. You’ll probably first look for an illustrator. Someone who has the style of art you desire. And then you’ll see if they also do graphic design. If they do, sweet. If not, no worries.
The main place to find an artist is on the website Board Game Geek. There’s a forum titled “Board Game Art and Graphic Design.” Artists looking for work will post their portfolios. You can also make a post describing what style of artwork you’re looking for and you'll receive submissions. I recommend Board Game Geek artists since they are already familiar with board gaming.