Should I use raw beets in my smoothie recipe?
It sounds like this a juice recipe, not a smoothie recipe. It may work if you put it in your blender, as I've seen recipes for smoothies with similar ingredients like this one but it will probably need some more liquid. 

When you juice, you definitely use raw fruits and vegetables but you need to run it through a juicer which will remove most of the solids, which is why you generally don't see any liquids added to the recipe.

We use raw beets in our juices all the time and, while they sometimes have a very earthy flavor, they do add a lot of sweetness. I've found the earthiness can be balanced with a little lemon juice.

... Anyway, if you want to make a smoothie with these ingredients (though I can't see the entire recipe), you can certainly follow the one I linked to earlier, which very clearly states that you use raw beets.

If you do want to follow this recipe, I can tell you that it pretty much is guaranteed to be calling for raw beets, simply because it doesn't say to cook them. Canned beets are generally already trimmed and peeled, if not chopped, so these steps don't make much sense to list.

While I wouldn't want to eat a raw beet like an apple, you don't need to cook it to consume it in a liquid form.