A proper musical would have involved dancing. Steve didn't dance. So, once he'd made it up to the balcony, he was left without a next move. (Okay. So the next move was pretty clear. He was just not sure how to go about it - to the best of his knowledge, 'want to go to dinner?' was not in any song in existence.)
So he reached out and offered his hand. Gestures would have to serve as stand-ins.