Tenant is complaining of mold in the bathroom, what do I have to do?
The bottom of the shower curtain is black and the corners of the tub are orange and brown. They are asking me to take care of this because the bathroom doesn't have vent in the ceiling. What should I do?
The only way this would be your responsibility is if you provided the shower curtain and it was already in bad shape. The fact that the bathroom isn't vented in the ceiling does not matter - there's a door. The mold collecting in the corner of a tub (as long as it wasn't already there) is part of the routine maintenance required of the tenant.
Make sure the tenant leaves the bathroom door open after using the shower/tub. Once a week, the tenant should spray the tub and curtain with any number of household products available at the grocery store.
Now, if this was a plumbing leak, excessive moisture coming in from the outside, a leaking a/c system in the attic, etc., the landlord is responsible.