Well, one reason to have some confidence in Ron Paul's integrity is that he can stand on his record. A record anyone with an ounce of motivation can access.
He has written volumes of op-eds and speeches that are available in the public record and, of course, online. His views are clearly stated and are consistent over the length of his public service. His congressional record is testament to his principled stance. His personal life exudes responsibility, intelligence and other admirable qualities.
Ron Paul isn't a "handled" politician. He does have advisors but they aren't chosen for their ego stroking or rabid mad dog win at all cost no matter what talents.
What you won't see Paul do is insult or resort to personal attack politics. He doesn't make jokes about other people's expensive haircuts, although he might if the taxpayers had to pay for it. He talks to the issues not the personalities.
In science, one can ascertain the properties of a material by careful observation over a long duration. Well, Ron Paul has had over 2 decades in the public record and his innate qualities are rock solid.