If you pay rent for use but not exclusive possession of the basement and outdoor space, then the dogs may go into those areas--in essence, in this scenario, those areas are "common space" which may be used by all residents. That said, if he is dog damages or destroys your property, he would be liable for the cost to repair or replace, and if he will not pay voluntarily, you could sue him.Also, even if the area is common space, if his dogs are making it unhealthy or unhygienic and continue to do so after you provide written notice to stop doing so, the landlord would be violating either or both of the implied warranty of habitability or the covenant of quiet enjoyment and you may have grounds to either sue him for compensation (e.g. an effective reduction in rent for the period this is going on, to compensate for the impairment in your use) and/or in extreme cases, even break the lease. If you think this is the case, speak with a landlord-tenant attorney.If you are paying for exclusive possession of some of these areas (e.g. you get 1/2 the basement as part of your rental--you have a defined area that is yours), then the landlord may NOT let his dogs enter that area. If he does, besides being liable for damage, he is violating the leasee, which again could give you grounds to sue for monetary compensation or even possibly terminate the rental.