What is the red tassel for that hangs down from the top of June’s Journey screen?

It is a Chinese tassel decoration with an ox symbol, part of the frame, in celebration of Lunar New Year of the Ox. It doesn't do anything else and it signals the start of daily gazebo rewards for 7 days.

It is just decoration. It is part of the design for the Lunar Celebration. If you look at the gazebo on the right side of your island next to the red plane and below the Wedding Chapel you can see the celebration. Each day you can earn a prize by playing rooms.