The secondary was a major focus for the Patriots entering free agency and the draft. That's the biggest weakness for what is a strong team. They can help their chances quite a bit by upgrading the secondary with players like Taylor.
SB Nation's Pats Pulpit explains the pick:
With the 62nd pick, I was really in a unique position from a Patriots perspective. With the first three picks, I addressed the front seven of the defense with the selections of Whitney Mercilus, Kendall Reyes, and Shea McClellin. That left the biggest Patriots need as safety. I considered going BPA (possibly WR Mohamed Sanu), but at the same time, safety is such a big need for the Patriots that I didn't think I could leave the position unaddressed. While it would have been ideal to get a safety such as Mark Barron or Harrison Smith with one of my previous picks, Barron was never available and Smith went right before the Patriots' last pick at 47 to the Jets. So at 62, that left three safeties remaining that I would consider: George Illoka, Markelle Martin, and Brandon Taylor.
None of these three safeties is anywhere close to being a finished product, but all are late 2nd-3rd round prospects who have some strengths and could become immediate contributors in the Patriots' secondary. Ultimately, I went with LSU's Brandon Taylor, who has risen up draft boards as of late and who the Patriots have shown some interest in. Taylor is a reliable safety who is both smart and hard working -- two characteristics the Patriots love in their defenders. He can be a force against the run in the box, is instinctive in coverage, and while has only average athleticism and coverage skills for the position, I think he could develop into a solid starter at strong safety for the Patriots (with Patrick Chung transitioning to free safety).