A Walk Score is essentially a rating of how walkable a neighborhood is, based on the density of amenities. The more dense a neighborhood is, the higher the walk score, usually. I know the University District is very walkable, so it probably has a really high walk score, overall.
Getting more to the safety issue: When I was personally choosing a neighborhood to live in, I wanted to be able to park my car or get off a bus and quickly be home (so no long walks home, alone, in the dark). That came down to the very specific buildings I was considering renting in. I chose one that had a well-lit entryway, covered off-street parking and was 1 block from a bus stop.
Another thing that makes me feel safe is being in a busy neighborhood where people are out and about, walking their dog, taking their kids to the park, getting a cup of coffee and whatnot, so I personally equated feeling safe with a neighborhood that was more walkable and lively. If you're into walkability, Fremont, Ballard, Greenlake, Phinney Ridge, Roosevelt/Maple Leaf and some parts of Queen Anne and Magnolia have that walkable feeling to me.
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