What's the best way to handle it when a civilian thanks you for your service?

I definitely have to agree with you there Mr Baker.  I always come right back with a "thank you for the support."  It is a great way to let them know that they are appreciated and also a good way to "shift" the awkward feeling that sometimes comes with that initial "thank you for your service..."  I honestly believe that everyone has a role to play in the defense of our great nation and its way of life.  Some of us serve, some of us have served and for whatever reason, some cannot or do not serve.  Those that cannot serve or do not serve sometimes express a form of service on other ways like thanking service men and women at airports, maybe purchasing a meal somewhere or giving a kind word of support.  That is their form of service to our nation and I am thankful for that.

With only 9% of all Americans having served a single day of duty, and only .05% staying an entire career...just say "Your Welcome". Be proud of your contribution and let them say thanks. They want to be sure and acknowledge your courage and service.

That's a good response. I'm going to have to steal it.

I learned that one from a crusty old salty dog I used to work for "Thank you for your support."  and I try my hardest to make sure that I slow down, shake their hand, and make eye contact so they can see my sincere appreciation...

I humbly say, "It's my pleasure."

Outstanding response - I will "borrow" those from you and use them.   

It is imperative that we are mindful that those
that stop and convey thank you for our service have someone in their family served. I think of the gold star
Mothers who have lost their Son's and Daughters and take the time to tell us
thank
you. It is vital that we show compassion and support.

nice!