A DBA is not a person can cannot sue in court. A DBA is simply a name that a person/entity uses for business other than the person's/entity's legal name.
The landlord would have to sue the tenant, and if the trust owns the property then the trust should also be the landlord and enter into the rental contract with the tenants. Then if there is a need to sue tenants, it would be the trust that would have to sue. The downside to that is that the trust must use an attorney to do it, you could not represent the trust yourselves. Although, even if you were the landlords yourselves it would still be a good idea to hire a lawyer to sue the tenants for you.