Seeing snow monkeys, the red faced furry monkey bathing in hot springs in the middle of winter, is a must stop attraction while travelling around Japan. But only assuming you're visiting at the right time. For us, we had already set aside dates for our vacation. Then when we planned to stop at Nagano for our first hot spring experience, setting aside a bit of time to visit Jigokudani Yaen-Koen for snow monkeys was a no-brainer.
Unfortunately for us, visiting during the first half of November was too early. Upon arriving, before the approximately one mile hike to the ticket booth + hot spring, there was a sign that clearly posted "No Snow Monkeys". Snow had not yet fallen, and the temperatures were still manageable with a light jacket. Based on the reports while we were visiting, our chances of seeing snow monkeys were small, just based on luck if the snow monkeys decide to come down. If there's enough food in the forest, there isn't a need for the snow monkeys to visit the hot springs.
From our experience, I would imagine the best time to visit is when it is snowing. So if your vacation group can tolerate hiking a mile through the snow, then December thru March will give you the best chance of seeing snow monkeys in action!
If you are hoping to get lucky (right before snow falls), the only upside to missing out on snow monkeys is it will be a more pleasant one mile hike and you may catch the red fall leaves of the trees. So either way, bring a camera to take pictures along the way.
If you're still in the planning stages of visiting Jigokudani Snow Monkey Park, then a good resource is to check out the live webcam (remember to take account timezone difference). You can either navigate to the webcam from the official Jigokudani site, or just google "Jigokudani live cam". The live cam is not high frames per second, but you'll get an idea of what to expect.
To get to the Jigokudani Snow Monkey Park, our hotel + onsen (Kokuya Hotel) provided a free shuttle to the park. They gave us instructions on how to either take the shuttle back to the hotel or train station (if we already checked out of the hotel). Because we had time to kill, we opted to walk the entire way to the train station (the hotel again dropped our bags off at the train station).