My impression from the picture was that the Opuntia was OK.. As Jeff says, it all depends whether that brown part is soft or not. There seemed to be bits of the Euphorbia which are OK too. Anything soft and squidgy needs cutting off as the others have described. But for anything that you are going to "wait and see", it would be a good idea to let the soil get completely dry. It's the combination of cold and damp which is the real killer. And cacti and succulents are designed to cope with prolonged drought, so you could easily leave them without water till April or May.

If no-one has said - the Euphorbia isn't actually a cactus - it's a succulent plant closely related to the Christmas Poinsettia. They're a fascinating group of plants, very varied in form and growth pattern, well worth growing.