According to the OED the word bailout was first used in 1930 to describe the actions of pilot jumping out of an airplane.

  1. to bale out. [Usually so spelt, as if the action were that of letting a bundle through a trapdoor; but also (esp. U.S.) as bail, as if a use of BAIL v.4, to lade out.] intr. (Of an airman) to make an emergency descent by parachute from his machine. Hence also (rare) n. bail-out. orig. U.S.

1930 C. J. V. MURPHY Parachute 272 Some say the pilot ‘bailed out’ the moment he went into the spin. 1932 N.Y. Times 11 Apr. 32 He successfully bailed out of an airplane at an elevation of 1,500 feet. 1939 F. D. TREDREY Pilot’s Summer 28 If you bale out and land in water..a smart rap will release the whole lot and you can swim free. 1940 Times 15 Aug. 42 He baled out before his machine crashed. 1955 Sci. News Let. 8 Jan. 23 The purpose..was to explore human tolerances during a high speed bailout from jet planes.

A bale was originally a package or bundle of goods.

My cursory search failed to find any economic citations, although that is no doubt changing as we speak.

The link to a bale of goods didn’t occur to me before I looked. I was thinking along the American lines of bailing out a boat that was sinking, i.e. lading out the water.