I can speak first hand here...... I marked no, please do not contact my current employer, and I typed below that box on the application something to the effect of "Please do not contact my current employer until employment is accepted". It was reviewed during one of my interviews and highlighted by the pilot conducting the interview, and I was given the impression that they were very sensitive to this and would not put me in a compromising position (and that it was no big deal at all). I was offered the job. When the background check was done (its done by an independent company), they verified my employment by asking me to fax in copies of pay statements, from random, staggered dates through out my employment term there. I imagine w2's would work fine too. Once it came close to my class date, and my training/ drug/alcohol records had to be pulled from my then employer, SWA PD asked that I send them a quick fax, authorizing them to contact my employer. This was not done until the day I resigned, approx 3-4 weeks prior to the start of class at SWA.
I cannot stress enough that the SWA PD were more than accommodating in this area, and in no way do I believe it will negatively affect your application. Just realize that at some time they're going to have to get those records pulled. I did also give them the name of another pilot that I had flown with at that employer. I believe you have to list a second contact if you have not flown with your 'supervisor'. Make sure this person can verify your employment, is completely in the loop on what is going on, and can give you a glowing recommendation.
I was listed as a reference for a friend of mine. Similar situation, we had worked together and I was to verify his employment, but we had never "flown" together. I was asked many questions about his personality, ability to get along with others, would I like to hang out with him on a 4 day etc etc.
Good luck
Here we go again.