Olde coax cables may work ok for tv service, but cause problems with higher speed services such as internet or some streaming services.
Unfortunately, the only solution is to run new cables presuming they are older. Usually an electrician can do this cheaper than the cable company. Also, I wouldn't trust the cable company drilling holes in my house.
Your responsibility to insure it is working. Make sure you do not allow a tenant to run any cable inside the unit. They are permitted to connect to jacks provided and no where else. You will need to inspect regularly to insure they have not stapled cables around your base boards causing damage.