Yes you can run a criminal background check on rental applicants but be very careful how you use the information as it’s very easy to violate fair housing laws. If you are going to run criminal background checks you must disclose that on your application and you must run the check on every applicant.
If the check shows criminal convictions you cannot reject every applicant or you will be in violation of fair housing laws. For example, you can deny housing to someone who has been convicted of manufacture of controlled substances but it is illegal under state and federal law to deny housing to someone because of past drug addiction as past drug addiction and alcoholism are classified as disabilities.
An alternative to running criminal background checks is to tighten up your other acceptance criteria. Recently incarcerated individuals most likely have red flags on their credit report, gaps in employment and less than wonderful landlord references.