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.

The Italian Chronicles article is an interesting contrast to what I read recently on http://www.savevernazza.com where they list the restaurants, businesses and accommodations in Vernazza, along with their anticipated opening dates.
Some opening dates are still uncertain, but many on the list say they’re opening in March and April.
I wouldn’t write off Vernazza in your travel plans to Italy. Vernazza needs our business. She may not be as beautiful as she was, but like a friend who isn’t well right now, go to visit her, especially this year.
Check http://www.savevernazza.com for the latest info from people who live there.
I’ve updated the post with the link to opening dates. It looks like a few will open in March with more hoping for April or May. Visitors should be aware that construction will still be going on if they visit in spring.
I’ll update my article re Vernazza too. Good to hear that some businesses will be reopening soon. I’d hate to put people off going there – but was only reporting what I’d been told by a tour operator in the area.
I have added a link to savevernazza so people can go check on the situation.
Thanks for reiterating that its business as normal at three of the five Cinque Terre – my fear is that people think all five of the Cinque Terre were affected by the floods – which is not the case.
All the best from Milan, Italy,
Alex