Welcome to the Wefugees community and thank you for your question.
Yes, it is possible to apply for asylum at certain airports. They need space for accommodation which is why it's not possible at every airport. In Germany for example, you can have your procedure at Berlin-Schönefeld, Düsseldorf, Frankfurt-Main, Hamburg and Munich.
I can't tell you about the procedure in other countries, but in Germany they will check within two days, whether your application will be processed or if it is "manifestly unfounded". Based on this decision you will either be allowed to enter Germany (and to have a procedure) or not. But you won't be sent back without authorities checking your application.
For further information about the procedure in Germany, please check out this website by BAMF: