The states now have a number of registry's that are online and searchable.
In my business (security sales and installations) I am required to provide a criminal records search for myself when I renew my license and any employees as well.
I go to my local courthouse and pay $5 and they search the records and give me a print out.
In the application to rent, the person should list the past 2 or 3 residences..... simply contact the local courthouse in that town and check the records there.
There are also sex offenders registry listings available. It would certainly be wise to check that as well. You don't want Chester the Molester living next door to a single mom with a couple of kids. They are required/supposed to register in any new jurisdiction BUT..... they often don't.
A guy who runs a small home improvement/repair/handyman service in this town..... lives 2 doors down from me, has lived here for 5 years or so, has worked in my house and on my other properties doing painting and repair, was just arrested 2 months ago..... failure to register as a sex offender. No one knew, but the state eventually tracked and found him and arrested him. The damage this guy could have done, had he been active would be catastrophic to the folks and the kids here.
check the online state and county registries.