Call ahead. Give them a week's notice that you'll bring a double bass on the flight, have the weight and length ready as well. Some airplanes, especially on shorter routes, have cargo holds that cannot fit a flight case through the door. When booking your flight, the website might be able to tell you what plane they use on each route, or just call the airline and explain your needs.
Oftentimes employees are not sure of their own policies for a bass, so being as well versed as possible before you get to the airport is valuable. If you've got a smart phone (or just print it out), show the airline/TSA agent what their website states if needed.
Check the bass, and expect an oversize baggage charge.