Will a reckless driving charge show up on a background check?

An arrest or conviction for any offense can show up on a background check.  And most states now share information as to DUI charges through the Department of Motor Vehicles.  I would suggest that you seek to have the records sealed or expunged if you meet the qualifications in your state.  Many criminal attorneys can help with this.  Once it is expunged you do not have to worry about it appearing on a background check and answering those questions on employment applications.  Just make sure that once the expungement is completed you do a little checking to make sure it was properly done.  Sometimes things slip through the cracks.  Good luck.