Generally no – Minnesota law provides that, unless specifically authorized by law, no employer or prospective employer may “require an employee or prospective employee to pay for expenses incurred in criminal or background checks, credit checks, or orientation.”  Minn. Stat. § 181.645.
The primary exception for the rule is for teachers.  Minnesota law allows a school hiring authority to require any individual who applies to work in a school to pay for the cost of his or her legally required background check with the Bureau of Criminal Apprehension.  Minn. Stat. § 123B.03, subd. 1.
Takeaway:  Unless specifically authorized by law, employers in Minnesota may not require applicants to pay for the costs of their background checks.