Cutthroat can be found from around Hope down to Sumas mountain - more or less. There is more accessible water on the south side.
The book Fishing Fever provides info of many access spots:
http://www.fishingfeverbc.ca/ I am not sure if it is in print but the webpage indicates you buy it on line directly via the website.
Google Earth, Google Maps are also useful tools. Be aware that many good looking access point on these will be through Private Property or First Nations Reserves. Respect these ask if possible or when in doubt find another place.
While most folks fish backwaters and sloughs - side channels are most often productive for me - they can be sizable and swift. However the key is cutthroat prefer slower water than rainbows. Current edges, tail outs, the wash behind large hunks of waterlogged woody debris (stumps and logs etc) and are all prime spots. Once you know this you can also find fish in the main part of the river.
However some perfect looking runs will seldom produce. If you find a consistent spot by next season after the early summer flood it may no longer be that productive but the prime water may simply have shifted 50 meters or so up or downstream.
Watch for rises. As it warms and insects & minnows get active there will often be some visible activity. Salmon fry are starting to appear. I saw some Wednesday when I was out. Within the next 6 weeks they will become the major prey item.
Most of the North Shore streams that enter directly into the Fraser will also produce some cutts. The Stave was mentioned the Harrison is another. Often these are better when the weather warms and the Fraser rises a bit and gets some color.