I never gave this question much thought. I guess you might call it a "full-service" building or an "elevator-outfitted building" but those are kind of awkward terms. If you want to know if a building has an elevator or not, you might just have to visit it.
I agree with the above post. Most properties, whether they are hotels or apartments, will specify if the room does not have elevator access, especially for people with potential handicap concerns. As with all property, I find that it is always best to go and see it for yourself.
I do not really believe that there is any specific word that is used to refer to a building that has an elevator. I think that it is generally assumed that a tall building has an elevator unless it is specifically identified as a walk up. Or they might say in their ad that it has an elevator. But there is no specific term that is used in the United States, at least.