IMO, a better tenant who has the potential to stay in your property for a longer period of time will turn out to a better financial decision. Consider the costs and loss of rental income associated with vacancies, repairs, evictions/legal actions resulting from a less qualified but possibly higher paying tenant. Also the time, money, and effort required to screen replacement tenants should be considered. This could be worth $100+ a month to you. It is to me. The phrase, "penny wise but pound foolish" comes to mind when I consider trading a less qualified tenant for a few extra bucks in rental income a month.
That being said, always perform a thorough and standardized background check on tenants, consisting of income/employment/rental history, current liabilities, and criminal/credit checks and keep all relevant documentation. Also verify all information given to you by the potential tenant. Having a consistent and legal system that you use to screen all tenants can offer you significant protection if a potential tenant feels that you broke Fair Housing Laws and illegally denied them. Good Luck.