During their first private meeting in the Golden Country, Julia admits to Winston Smith that there was a faint, unorthodox look in his face that indicated he was against the Party, which is why she took the risk of slipping him a note. Winston discovers that Julia is a veteran when it comes to alluding Big Brother and secretly sleeping with Party members. Julia has developed a keen eye for political dissidents and trusts her intuition when it comes to seeking out allies, who are willing to carry on affairs in secret. Julia's act of slipping a private note to Winston is dangerous because it is against government policy and could result in her arrest. In Oceania, the government controls virtually every aspect of society and has eliminated individual freedoms. Julia's act of attempting to have an affair with Winston is considered a crime in Oceania, which is why she takes extreme precaution and inconspicuously slips him the note.

I believe that Julia writes that note to Winston because she senses in him a kindred soul.  What I mean by this is that she sees him as someone else who hates the Party and what it stands for.

The note is so dangerous because people in that society are not supposed to love each other.  Julia herself is part of the Junior Anti-Sex League, which is meant, in part, to keep people from forming romantic attachments.  This is meant to keep the people from being attached to anything but the Party and Big Brother.  So trying to become emotionally intimate with someone is really subversive, it would undermine the society if people did that.