The landlord of a rental property has a duty to maintain the property in a reasonably safe condition. As a tenant, if you see a dangerous condition on the property, you should contact the landlord, prefurably in writing and report the complaint. If you do not receive a response you should then contact Code Enforement in your city/town.
In the event that you are injured as a result of a dangerous condition you should immediately take photographs of the the condition.
It should be noted that the landlord’s insurance may have medical payments coverage that pay your medical expenses relaterd to the fall. If contacted by said insurance carrier you should not give a recorded statement without the presence of an attorney.http://www.thelawshack.com/nysb/Landlord_Tenant.html