That's going to be tough in the current environment...especially the reckless. You can pull all of your national and state driver records in the hopes that they don't all show up, but odds are that anything that recent will show. If it happened in 1982 before the computers were quire as good, you might have lucked out and had it slip through the cracks but nowdays...not so much.

If you omit it from your application, you might get hired but you will quickly get "unhired" when they find it on your background check. Also they normally do background checks (which are expensive) AFTER you report for training. This means that you will have been fired from a 121 airline job, and will probably never work in the industry again.

How long ago was the reckless? If it was 15 years ago, and then just a couple tickets since you might be OK. But if it was within 3-5 years you will probably need to work in 135, corporate, or other 91 stuff to put a few years between you and the reckless. You will also need a PERFECT record from now on, not even a ten-over. I can afford a ticket, but you cannot.

Hopefully you were very young when it happened. If you were 25+ plus they will have reservations. If you were over 30, they will doubt that you can change since most people don't change much after 30.