That depends a good deal on just where you are fishing. Generally speaking, the South Sound and Hood Canal provide a longer salt water season than the North Sound. In the larger North Sound rivers cutthroat begin entering the rivers as early as July though they'll continue to trickle in through the winter, becoming less available in the salt.

Due to the generally smaller rivers and streams in the South Sound and Hood Canal cutthroat tend to spend a longer period of time in salt water (Willapa Bay and possibly Grays Harbor as well), sometimes not entering their spawning streams until as late as December. Between late-entry fish and those who have spawned early and returned to salt water there can be a near-year-round fishery in these areas.

Taking these things into account, the best time to look for them in the salt is probably in the late summer and fall. Of course there are always exceptions, I've caught cutthroat off the beach at Golden Gardens, in north Seattle, in January. Keep in mind that coastal cutthroat, sea-runs in particular, are probably less-studied than any of our other salmonids (because they have no commercial value and, until recently, have not generated a lot of angler interest) and, at the same time, exhibit a wider variety of life histories than any of our other native species with the possible exception of the bull trout.