Although last year's floods caused severe damage in two of the Cinque Terre villages, the other three remain open with no problems. Many services in Monterosso are already reopening and most should be open by spring, although Vernazza currently remains closed, according to this Italy Chronicles article, where you'll also find information about how to check for updates. Some Vernazza businesses anticipate opening in spring, you can check the list on the Save Vernazza Travel Advisory.
Three of the four number 2 or blue trails, the main hiking trails connecting the five villages, remain closed. You can check to see if they're open on the Cinque Terre National Park site (although it's in Italian, the green dot indicates open and red means closed). There are also other trails in the area that can be accessed, such as red trail number 6.