Ahh - I didn't get the point that you don't care where you end up, so long as you go
somewhere.
Also, DS normally involves
alternating climbs and dives through the shear zone, not what I would call a sustained climb. But I see where you're going (aerodynamically, if not geographically).
Anyway, thus far a manned sailplane has managed to maintain peak altitude (350m/1100 feet) and gain
some headway against the wind, by DS with and into a
40-knot shear at 1000 feet/300 meters. (42:1 glide ratio, 158 kt Vne, MGW 990 lbs, PIK-20). And maintain peak altitude (but drifting with the wind) with a different craft (glide ratio 39:1). I suspect with a manned craft, the prime limiting factor of what can be achieved will be the Vne (pop up into a shear of 70 knots with an airspeed already at 140 knots, and the wings come off). As always in aviation there is a trade-off in strength vs. weight (which will also affect glide ratio). A specially-designed craft may be necessary. Model gliders don't have to house a non-lift-producing human body, and can devote that weight to extra structural strength, and birds have nerve sensors (pain!) to warn of overspeeds.
For jet streams, you will need 1) oxygen (already mentioned - another weight penalty) and 2) a way of getting up to jet altitudes (on the order of 33000ft/10000m) - either a tow (what tow craft can climb that high at a low enough speed for safety - U-2?), or a preliminary climb in mountain waves (the traditional lift source for sailplane distance records) - perhaps a climb along the Andes wave to catch the southern subtropical jet over Argentina/Chile somewhere. GPS for location plus real-time data on the jet location as it meanders might be needed also.
I guess the first sentences in my second paragraph is as close as I can get to answering your basic question.
(Thought experiment - since jet streams have a core, and thus lateral shear as well as vertical shear (cross them sideways, and the wind speed increases into the core, and then decreases exiting the stream) - what happens if you "tack" back and forth through a jet stream, with or without vertical motion as well, at an angle to the wind so that you get to use some of its force? Or an even more complicated path of climbing, diving and tacking all at once?)