I know someone very closely involved with building the station. However I can't remember who paid - I think it was a combination of the council (who own the airport) and possibly the TOCs. Bear in mind the deal was done in the heady days of Railtrack, and at the time when Midland Mainline (as the TOC was then) was starting some ambitious growth proposals. I think it relied on the doubling of the MML service to have one of them stopping there. I do remember the funding agreement taking an age to resolve.
I also remember that it cost about £25m in late 90s prices, or over £40m in today's money.
It was certainly tied in with the airport's growth plan to take passenger numbers up to around 10million. The new terminal building was open by January 1999, and that is when the airport took off if you'll pardon the pun.
Luton has seen spectacular growth in the last 3 years, which given that it has been a building site for most of it is nothing short of remarkable. When the March 2017 stats come out in 4 weeks it will show that 15million people went through the airport in the previous 12 months. When the terminal extension is finished, and the light rail connection done, plus the improved connections through Thameslink, it will easily go through the 20m barrier.
Assuming you are thinking about LBA, a station would be cheaper, but the airport would need to be ambitious about its growth - current plans of c7mppa by 2030 probably aren't enough. It probably wouldn't be much use as a conventional parkway, except for people from Yeadon. The Luton Parkway concept was based on people getting off the M1 earlier to get into central London, rather than driving all the way in, or to Stanmore / Barnet etc for the tube.