Once an industrial area, many of the old factories that were the heart of this neighborhood have been repurposed. As the manufacturing companies began to move out in the 1950s, struggling artists looking for affordable places to set up shop began to move into the area. As more artists moved into the area, businesses such as restaurants and clubs began to move in as well and today, it is now home to a wide assortment of art galleries as well. There are plenty of amenities for you as well as a dog park and a dog hotel. You will also find a pet store and a vet just to the west of the district.
Los Angeles is warm all year round, so you will have more trouble from the heat than just about anything else when it comes to walking. There may be some rain in the winter, but for the most part, you can walk through the season.
The Art District is located in downtown Los Angeles and as such, the terrain is very much an urban style. What this means for dog walkers is that you will be walking entirely on concrete here unless you find a park or greenway where there is some grass. Given the frequent hot temperatures, you may want to consider early morning walks since temperatures on the concrete are sure to frequently be hot enough cause discomfort to paws.
The elevations here are quite flat, so it serves to be extra cautious. This area can be a lot of fun to wander with the colorful artwork and plethora of shops, galleries, and outside eateries and it might be tempting to stay out on the concrete for much longer than is good for your dog's feet. On the positive side, when the temperatures are not too hot for a long walk, the lack of hills makes it much easier to cover a lot of ground.
Unfortunately, because of its location in the middle of downtown Los Angeles, there just is not a lot of greenways close by and certainly none within the Artists District itself. As much fun as it is for you to wander through this district if your pup is really craving a bit of green to sink their feet into, you are going to have to hop in the car to find a park or greenway to satisfy that craving.
There are a lot of small parks in the neighboring communities to the west, the closest of which is Hollenbeck Park just on the other side of 101, I-5, and I-10. There is a fair amount of grass, some walking paths, and even a small lake, enough to make the trip worth it!
To the north, you will find a larger area, Los Angeles State Historic Park and Elysian Park, where there is several trails, some hills, and some nice gardens to explore.