Of course, the easiest way to be seen as honest is simply to be so. Some folks will notice and some won't, although as we all know, they will certainly notice - sooner or later - when your department, agency, or company isn't being honest.
What if you want your honesty to be conspicuous, though, so that it readily becomes an accepted part of how you're seen and branded? Start by thinking hard about the specific actions in which your organization engages that represent building and maintaining trust. (Remember, simply saying you're honest means nothing - everyone hopefully expects that already. Instead, again, think about specific behaviors that demonstrate both your honesty and reliability.) Once done, put those actions forward explicitly in your conversations, your promotional materials, and of course, in every piece of training and supervision you provide so that every employee unerringly engages in those behaviors with each and every one of your customers.
Does such conspicuous honesty really matter? There are plenty of studies to provide facts and figures about the damage caused by a lack of customer trust but the simplest test is this: would you rather deal with a department, agency, or company you trust or one where you either aren't sure or already know that they can't be trusted? It's tough to imagine that the former wouldn't win every time.