I think they'll have to wait and see how 2016 goes. I think the long-term solution is Rafael Devers, who's 19 this year. He's the #13 prospect in baseball and a lot of scouts don't think he'll be able to stay at third and that the DH spot will be a nice fit for him. If not, Josh Ockimey and Sam Travis are also pretty solid prospects that have a chance to be major league starters and would fit will at DH.
But then you need someone to hold that spot for the two or so seasons until Devers is ready, and I think Boston has a lot of options.
Right now, they plan on converting Hanley to a first baseman. Travis Shaw, however, is an excellent defender and really impressed with his bat last season. If Hanley can rebound, him and Shaw could make a nice DH/1B combo.
If not, Sam Travis should be close to major league ready by the end of the season. If Shaw continues to hit, he can hold down first base and Boston could slot Travis in the DH spot.
However, those are both risky situations. There's no guarantee that Shaw continues to hit, Hanley's the farthest thing from a sure bet, and Travis still hasn't played above AA yet.
So if they do end up signing somebody, there are a few names that stick out. I don't see them spending big money on a past-prime player with injury risks that will still require pricey contracts (i.e. Carlos Beltran, Mark Teixeira, etc.). There are two big names - Encarnacion and Bautista - but their age and cost would deter Boston, in my opinion. I think they'll go for a guy they can sign to a 3 year or less contract for no more than $12 million AAV a year. That group could include Brandon Moss, Adam Lind, Adam LaRoche, or Justin Smoak. Mitch Moreland, Mark Trumbo, and Kendrys Morales fit the bill as well, but they'd be more costly. Reddick would be a great addition and the front office likes him, but he would probably require a longer contract, and you lose some of his value as a DH because he's above-average defensively.
So to answer your question with more certainty, I think we need to see how this season goes.