Can "Stealth Tower" copy your opponent's bases?
Your opponents bases are In Play so Stealth Tower can copy them too.

After Stealth Tower copies a base, it remains a copy of that base regardless of what happens to the original (destroyed or returned to hand, etc).

This relies on an understanding of what "In Play" means in the context of this game.


  A Player Turn consists of three phases: Main, Discard and Draw.
  
  1) Main Phase
  
  At any time during your Main Phase you may perform any of the
  following actions, in any order you wish.
  
  
  Play cards from your hand
  Use the Primary Abilities of in-play Bases
  Use the Ally/Scrap abilities of in-play Ships and Bases
  Use Trade to acquire new cards from the Trade Row
  Use Combat to attack an opponent and/or their Bases
  
  
  Playing Cards: There is no cost to play a card, simply place it in-play (face-up on the table in front of you [the active player]).
  There are two types of cards you can have in your deck: Ships and
  Bases.
  
  Bases: Unlike Ships, Bases are not discarded at the end of the turn.
  Instead, they remain in-play until they are destroyed or scrapped.
  
  2) Discard Phase
  
  
  Lose any Trade remaining in your Trade Pool
  Lose any Combat remaining in your Combat Pool
  Put all in-play Ships into your Discard Pile [and leave all Bases in-play]
  Put any cards left in your hand into your Discard Pile
  
  
  3) Draw Phase
  
  Draw five cards, then your turn ends


In the Main Phase the first action says you can play cards from your hand. Playing a card involves putting the card from your hand in-play in front of you. During the Discard Phase all in-play Ships are discarded, but Bases (and Outposts, which are Bases) remain in-play. Therefore Bases Played by any player are in-play, even after their turn ends.