APL (Dyalog Classic), 13 bytes

⍪,⊃∘.,/⎕,¨⊂⊂⍬

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output:

 1 2 3 
 1 2   
 1 3   
 1     
 2 3   
 2     
 3     
       

there's a blank line at the end to represent the empty set

APL (Dyalog Classic), 13 bytes

⍪,⊃∘.,/⎕,¨⊂⊂⍬

Try it online!

Output:

 1 2 3 
 1 2   
 1 3   
 1     
 2 3   
 2     
 3     
       

There's a blank line at the end to represent the empty set.

Explanation:

⎕ evaluated input

⎕,¨⊂⊂⍬ append an empty numeric list after each element

∘., Cartesian product

/ reduction (foldr)

⊃ disclose (necessary after reduction in APL)

At this point the result is an n-dimensional 2-by-...-by-2 array, where n is the length of the input.

, flatten into a vector

⍪ turn the vector into an upright 2ⁿ-by-1 matrix, so each subset is on a separate line