I read on this answer by Anish Sheela:
Sign indemnity form saying that you accept personal responsibility to pay the fine imposed to the airline, in case you are denied.
Which airlines offer indemnity forms? Do all airlines have it, or just a few ones?
From this answer by Zach Lipton, at least Qatar Airways has it:
All airlines should have the indemnity forms. But they will allow you to use that only if they are reasonably sure that you will be admitted to the destination country. Airline will have to pay fine, whether you sign this or not. They will strongly try to recover from you any losses they incur, as they have documentation that you agree to pay.
An example would be the passport is slightly damaged, but it won't be usually a problem, airline could offer the form to cover the risk. Another example would be if you are trying to board to USA without a visa, no amount of convincing will get you this form.
Thai AirAsia has it (at least in 2006). Example from https://www.flickr.com/photos/zhaffsky/352652268:
