I was in Hakone last year early January (right after the ekiden road race) and everything was open e.g. ropeway, cablecar, boats etc. Weather was good though. While it wasn't super crowded, there were quite a lot of people there as well. Also IIRC wasn't that cold - I thought it was just nice with all the walking that we did (Owakudani, the Tokugawa-era checkpoint route etc)