See attached image. These tiles are in the common area of my apartment building. As you can see, there is significant cracking. My building was built in the 60's, I know that the 9-inch tiles in my apartment contain asbestos, and I suspect these 12-inch tiles in the common area also contain asbestos. The dark area on the right side of the image is stain from where a rug normally is.
Asbestos is only dangerous when it becomes airborne, which happens when whatever contains it is turned into dust. For example it would be a very bad idea to remove these tiles by breaking them into bits with a rotary hammer, and even worse to cut them with a diamond disc grinder. The mortar/glue below may also contain asbestos fiber as reinforcement.
However in their current state, they are not dangerous, as the asbestos fibers won't become airborne.
If the building contains asbestos though, you might want to check the fireproofing and/or insulation. These are the most dangerous, as they are spray-on fibers and tend to degrade with time, releasing the asbestos into the air.
It depends on how dangerous you think asbestos is. I will disagree with the previous answer.
I would not let anyone other than myself walk on them if I were fixing a home. In fact I walked on asbestos tiles in this condition at a home showing and roasted the selling agent because of it.
You can see the cracks and chips on the edges of the tile. While walking these can become dislodged and airborne. Will it be significant enough to cause damage to you? Probably not. Is it worth chancing. Hell no.
This is actually a pretty easy fix for the super, nice quick sheet of glue or poly and lay tile on top. I would send a note to your super asking if the tiles where tested for asbestos and show them pictures of the damage. Although tiles are a big source of asbestos health issue why should you be paying for the chance to get cancer? Kicking a part of that tile that is cracked and dislodged would definitely push particles in the air.
As noted in comments in another answer - remedy could be as easy as posting warning signs in building about tile and calling city hall.
I agree with DMoore. If you have small children who crawl, then I would definitely recommend getting a building petition to see if you can't get the landlord to hire a licensed asbestos abatement professional and have it removed. Alternatively, he "might" be able to lay down new tiles over the old. It's a fairly cheap solution, safer than removing the tiles, and will solve the problem. Also check with your state's Real Estate Commission; there may be a requirement or procedure you can go through if the landlord does not respond/agree that the tiles should be removed or covered.
There are federal disclosure laws for lead (EPA) and asbestos (OSHA), so if the landlord didn't disclose these things to you, and provide you with a form that you had to sign that stated he did disclose them to you, then you have an complaint or possible cause to break your lease. Consult an attorney if you decide to do this! It's my understanding that as long as the asbestos is contained and intact, it's not hazardous. But, the key point here is that it is no longer intact. I am a licensed professional Realtor® in the State of Texas so your state will/may be different. This is only my opinion,and you should seek information from contacts and the agencies which govern these concerns.