Accessing Karlsruhe from a major airport

It is probably worthwhile to look into getting a Rail&Fly ticket. The general ideas are as follows:

• They must be booked through your airline in conjunction with an international flight to/from Germany. (There are some exceptions for some flights to/from European cities that are also served by Deutsche Bahn.) Not all airlines offer this service but most non-discount airlines that aren’t KLM/Delta that offer service to Germany provide this option.
• They cost 29€ Per person per direction for 2nd class. (Double the price for first, optional seat reservations are extra.)
• They are valid on any train including ICE trains.
• After a flight, they are valid on the calendar date of the flight and the following calendar day. Before a flight, they are valid on the day of the flight or the proceeding day.

Unless you are travelling a short distance, these tickets offer a very cheap price with incredibly huge flexibility so even if things go incredibly wrong with your flight, you don’t have to worry about losing the value of your purchased train ticket.

They also allow you to opt for the airport which has the cheapest/most direct flights (which is usually Frankfurt). Most major German airports have built in Train stations and since Germany has high speed rail links (up to 300 km/h).

Germans generally have good English, especially in industries which often interact with foreigners such as rail and air transport.

Definitely Frankfurt since the train station is in the airport. The airport actually has two train stations: one for regional trains (S-Bahn) and for regular long distance trains (Fernbahnhof). You definitely want the second one (Fernbahnhof). Every two hours there is direct train, that doesn't require changes. Tickets can bought at www.bahn.de and it's substantially cheaper if you buy them two weeks or so ahead of time.

There are currently only two airlines that go non-stop from SEA to FRA which are Lufthansa and Condor. Lufthansa will run you around $1300, Condor can be as little as half of that but they don't fly every day, so you have to have some flexibility in the dates. "normal" flights with a single layover should be around $1000. All prices are mid week, mid February.

A note on German train booking, if for some reason you are not using rail&fly (whose availability and price is dependent on the flight carrier you choose): you get two prices for each train, "fixed" and "flexible". Fixed (which is a condition of almost everything offered as "savings fare") means that you are buying a ticket for a given train and if you miss that train for whatever reason (except by fault of the Deutsche Bahn itself) you have to buy a new ticket to travel later. The Flexiticket is valid for any train travelling that day, and in connection of the uncertainty of intercontinental flights and the stress of long travel to an unfamiliar destination, I would consider it money well invested.

As for booking from home, you can buy the train ticket under bahn.de, payment by credit card is accepted, and you are obliged to carry with you a printout of the ticket. As said above, compare prices with Rail and Fly first - it seems that you will be paying around 40 Euros from Frankfurt with the ICE. This is the price for one direction, you don't get discounts for booking the return journey at the same time.

For the preferred order of airports: the other answers already explained why Frankfurt is by far the best choice. If for some reason you want to compare to second-best alternatives, then Stuttgart (1-2h), Strasbourg (1-2h), Paris (2-3h) and Zürich (3-4h) would come into consideration, in order of reduced time spent on board of a train. For basically all of those, you can find connections with only one layover somewhere other than Frankfurt, so if the options you have through Frankfurt are very inconvenient for you, you can shop around to see what happens if you fly to one of those. Smaller airports which are physically closer are unlikely to be a better choice because of more layovers or worse train connection.