MacBook Air 2013 battery drained by Dropbox

Make sure there are no processes hogging your CPU (this eats battery).

Open Activity Monitor from /Applications/Utilities.
Set the Show menu at the top right to All Processes
Click the %CPU column head so the triangle faces down.
Which processes are at the top? How much are they using?

This may reveal what's going on.

Also, you can check the health of your battery to make sure it's not a hardware problem.
Download the free coconutBattery app and see what it has to say about your batter's capacity.

(this answer was written for the previous edition of the question)

I didn't find any "most robust and lasting" choice yet, but I contacted iFixit re their replacement battery, and this is what I found out.

1. They claim that they sell the battery:

   • manufactured by the same company that manufactures Apple's battery
   • its initial power-on time on a single charge is better compared to the original battery: "buying a brand new battery will get you back to where your computer was [in same time between charges] when you first bought it"
   • there's 6-month warranty for a case if it performs poorly at customer's computer

2. However:

   • I failed to get any comment on how it can be possible that Apple allows its manufacturer to sell parts that Apple itself doesn't normally sell (which is the case for MBA-2011 batteries)

   • the sales rep couldn't provide any tests proving the "with better life" statement, nor even confirm that such tests were ever performed: "Unfortunately I do not have any tests to show [better life / better time between charges] as we have found those to not be needed"

   • could not confirm that any part of these statements are true:

     • the replacement will last comparable number of recharge cycles as the battery installed by an official Apple service
     • the replacement's poweron time on a single charge will be the same (or ideally better) than how could original battery installed by Apple service perform, at (most) every point in the battery life

The above data doesn't exclude what I experienced with low-quality third-party batteries for non-Apple products: initially they start with unprecedented time on a single charge, but after few months time on a charge deteriorates to less than an hour.

But let's see what answers a question on performance of iFixit batteries will bring.