Hi John
Paphos is obviously a lot quieter in January/the winter months but the resort is not closed.
Some of the hotels close for a couple of months and more have been doing so in the last 2 years. This is because the "package holiday" trade has suffered with more people booking their own flights/accommodation so some hotels have decided to close for the winter months rather than stay open.
You will be able to find hotels that are open but you may have to look a little harder. if you're looking for something special, try one of the spa hotels in Paphos on the 5 star hotel page like the Thalassa or Elysium (they should be open as not "package hotels"). Most restaurants stay open but may close for a day or two per week.
What I would say is that you are more likely to get rain in January than later in the spring. So, depending on whether you are happy to chill out in a smart hotel versus getting some guaranteed sunshine...even if it does rain, the temperature is still warm. We walked today (Dec 10th) in 20 degree sunshine.
Enjoy
Helen