This workflow allows you to invoke AirDrop in Alfred by typing airdrop.
Compared to hackish solutions that require AppleScript such as this and this, this simply executes airdrop.app directly, giving you the same experience as what you'd get when you type airdrop in Spotlight, e.g. it won't create duplicates if you already have one or more AirDrop windows or tabs opened, and lauching AirDrop won't bring all Finder windows to the front.
I'm actually a bit curious about why Alfred couldn't find airdrop.app after I added /System/Library/CoreServices/Finder.app/Contents/Applications/ to search scope. Until that works in Alfred, this is probably the best solution.
Download the workflow at Packal!