Yes the Times Square Ball does stay up all year so you can see the Times Square Ball year round when you visit New York City.
However the Times Square Ball only drops on New Years eve starting at 11:59 PM and takes 60 seconds to completely fall down from the top of the pole which then makes it to the ground level at 12:00 AM on New Years day.
The Times Square Ball is mounted on a special flag like pole that allows the ball to drop and then go back up.
December 31st 1907 is when the first Times Square ball drop happened to ring in the New Year of 1908 so it's been going on for a long time.
That means the ball drop has been ringing in 112 New years which is amazing and it will continue for many more years.