I'm on the Burke everyday, betw Lake Forest Park and the UW, for 7+ yrs now. In warmer nicer weather I'll commute down 35th or Sand Point, but the Burke is nice in the winter/rainy months. Peds are all inside, and most of the commute trail traffic has gone the way of the SUV until it gets sunny and warm again. I see maybe 10 people on the way in at 6-7 am, and even fewer after 6pm in the evening.
I've only ever seen cops on the trail in LFP. Usually they sit at the 165th intersection. A couple fat bike cops, sometimes a motorcycle cop and maybe even a squad car. Usually they're looking for cyclists blowing through the (ridiculous) stop signs on the trail in this area. They'll get you for "failure to yield" which I've heard is a $65+ fine and goes on your driving record. Never been caught: they only come out on nice warm sunny days, usually during the evening commute (4-5 pm). I have absolutely never seen them out if there's even a hint of precip. They may be there on weekends--I wouldn't know: I avoid the B-G and Sam Riv trails like the plague on the weekend.
Don't get me started on B-G stories...