"Did management somewhere get confused and think professional pilots are prostitutes?"
As chperplt said, this application fee thing is nothing new... I for one had to pay for the privilege of chucking my resume at American Eagle back in '99... And anyone who's familiar with the now-defunct AEPS knows quite well about paying to put your name in the pile. An old friend of mine had about $300 spent on application fees to the majors back in the late 90's, and for the record, didn't end up with even an interview with any of them for all his outlay.
In the case of Shuttle America, the fee is actually $30, not $38, and none of it actually goes to the company. It is paying for a service that screens the resumes before they get sent to the HR department for consideration... And it gives you the opportunity to complete the application online. I haven't actually SEEN this for myself, but this is what I understand from asking the CP. As you can imagine, we receive hundreds of resumes per month and it just became too much to deal with in HR. Very simply it reduces the mountain of paperwork and the service they are using only allows through applicants' resumes/apps that meet the minimums or are otherwise qualified, and THEN they can decide who to interview.
Good news is that if you don't meet the minimum qualifications, you won't be charged even if you attempt to apply.
In any case, I guess you just have to ask yourself if the $30 is worth the gamble. If not? Then don't apply. It's that simple.