Oh, they'd hit the arrest record before they even got there. The type of people I imagine I'd come across dealing with things needing a security clearance won't find those stories nearly as funny as I do.
It kind of depends on what the arrest was for.
One chap I met admitted to me he had in his past dealt dope, yet hadn't been afoul of the law. Though this was known, it didn't seem to preclude his getting a clearance.
MICE is the acronym the folks who do background checks worry about.
Money - Are you in financial trouble.
Ideology - Are you a communist or sum such?
Compromise - Can you be blackmailed
Ego - One so superior that the rules don't apply to them.