An aviation degree is certainly not necessary, and may be an expensive waste of time.

Pilot employers hire based on the certificates and experience you have, they simply don't care where you learned to fly unless it was a military program.

A carefully selected non-aviation degree can provide a crucial backup job/career in the event that aviation does not work out, or if you are out of work temporarily due to medical, furlough, etc. A challenging degree such as science or engineering may help you get hired at some airlines, since it shows you can study and are goal oriented.

Good pilot jobs are always competitive, bad jobs are only competitive when the economy is slow. Things that help...

College degree (4 year)
Good grades
No record of flight training failures.
No criminal record.
High flight time.
Military flight experience.
Pilot-in-command or turbojet/turboprop airplanes (Turbine PIC).
Time in larger airplanes.
Airline experience.
No aircraft accidents/incidents.
No FAA violations.

It is also good to have attention getters: Things which help you stand out. This could be things like college athletics, awards, really high GPA, unusual aviation activities, significant community service, even interesting hobbies. If you're just another slacker with a 2.6 GPA who plays a lot of video games and never does anything worth noting...get in line behind all the others.