During the famous balcony scene, there actually are not any stage directions indicating that Romeo climbs onto Juliet's balcony. However, later, while he is arranging for their wedding, Romeo does ask Juliet's nurse for assistance to climb onto Juliet's balcony and into her window.
We can tell that in Act 2, Scene 2 Romeo actually remains in the garden and does not climb onto her balcony because twice Shakespeare describes Juliet as entering onstage from above Romeo. The first time Shakespeare gives this stage direction is just after Romeo's very first line, and the stage direction is "Enter Juliet above at a window" (1). Therefore, during the course of all of the lines delivered, Romeo remains on the ground, in the garden, while Juliet is positioned above him at her window. Even the first time Juliet returns to him after going back into her room because her nurse is calling her, Juliet is still described as being above Romeo, as we see from the second stage direction, "Enter Juliet above" (147).
However, Juliet makes arrangements to send a servant to Romeo the next morning to inquire about his plans for marriage, if he is serious about marriage. Juliet sends her nurse to meet Romeo the next morning at 9 o'clock, and this meeting takes place in Act 2, Scene 4. During the meeting, Romeo not only tells her to instruct Juliet to go to Friar Laurence's cell for confession where they would be married, he also tells Nurse to later meet a servant of his behind the abbey wall. There, the servant will give Nurse a rope ladder, and that's what Romeo will use to climb Juliet's balcony and enter her room for their secret wedding night, as we see in Romeo's lines:
And stay, good nurse, behind the abbey wall.
Within this hour my man shall be with thee
And bring thee cords made like a tackled stair. (II.iv.173-75)
The word "cords" refers to ropes, while "tackled stair" can be translated as "rope ladder," showing us that Romeo plans to use this rope ladder to climb into Juliet's balcony and bedroom (eNotes).