Girls get hit on *all* the time in those situations: coffee barristas, sales clerks, flight attendants, waitresses... its hard to know if a guy is actually serious or just flirting.
So, somehow you need to get the point across you are serious but keeping in mind she might either not be interested, or might be seeing someone already.
How old are you? Are you a student or are you working? If you are working, you might consider going back at a time that isn't too busy and ask her out. Be sincere. Then leave your b-card w/a contact number or email. If she is interested, she can always contact you.
Whatever you do, you'll have to leave the ball in her court, so don't get your hopes up.
Second thoughts can generally be amended with judicious action; injudicious actions can seldom be recovered with second thoughts.
--Cyteen by C.J.Cherryh