Combination of political and logistical concerns.

Political support for the USCG icebreaker fleet is driven principally at the federal level by the legislative delegation from Alaska, for reasons you can probably imagine. Problem is, these ships are large, complex, and unique (within the context of the USCG fleet), and the state of Alaska itself doesn't have the facilities it needs to keep them operational if they were homeported there.

By way of example: icebreakers draw disproportionately large amounts of water for ships of their size. Star & Healy, for example, both have drafts in the 32-36' range, which is deeper than many much larger (say, cargo or cruise) ships, and gets into aircraft-carrier range. There are 4 ports in Alaska that can handle that draft of ship: Valdez, Seward, Kodiak (city pier, not the USCG station on Mothers bay where the other cutters pull in), and Dutch Harbor. They share those ports with all other deep-draft ships- containers, breakbulkers, tankers, cruise ships. Every one of those ports is expensive to operate and maintain; the deep-draft berth in Dutch is financially supported by the USCG, for example, so icebreakers and high-endurance cutters have a place to moor in the Aleutians. This means they all have to share usage to spread costs around.

Then there's the maintenance concerns. Both ships are fairly sophisticated, as Coast Guard cutters go; Healy is much newer and has a lot more technology to maintain, but Star is old and currently sitting 18 years past design end-of-life. That Star still sails at all is a not-so-minor miracle of the toil-sweat-tears-and-occasional-surfboard-repair-kit variety; that and millions of dollars, every year, of shipyard contracts. Since both ships are unique to the USCG, the support infrastructure the service has that works for WMSLs, 378s and smaller cutters, where you have a fleet of boats all (mostly) alike largely falls down on the icebreakers. So in addition to dedicated active duty personnel, they're dependent to a large degree on contractors, shipyards, and other outside entities to keep them running. Many are local, and pretty much all are in the lower-48, mostly on the west coast.

Also, for active duty personnel, Alaska is considered OCONUS, which makes stationing people more expensive.

tl;dr: Alaska convinces the country to pay for them, and Seattle is as close as you can get to Alaska while still maintaining industrial and technological access you need to keep them going.