PUZZLEBOXX

You could use the balloon as a kind of glove and then pick up the mug.  (But don't you think that would be cheating?)

You could place the balloon under the mug then gently inflate the balloon so that the mug is lifted.  (But how will you place the balloon under the mug if you cannot touch the mug?  And won't the mug just tip over rather than be lifted?)

The best solution is to dangle the balloon inside the mug and then inflate it so that it grips the inside of the mug - thus making it easy to lift.