As far as crabbing goes, life is pretty dry without a boat. If you can beg, borrow or steal your way into being invited onboard a boat, you'll do a lot better than you will from shore or a pier.
But, if you're unable to get out on the water, there are some piers around that are better than nothing. I wouldn't eat anything out of Elliott Bay, you need to leave the city for this.
To the south, there is a good public fishing pier in Des Moines. To the North, Taylor Street Pier in Bellingham offers some good over-water access as well. I'm sure there are other points in between, but I see people crabbing on these two frequently.
You'll also need a pot or a ring. They are different. A Pot goes in the water, and stays in, for hours or days at a time. It traps the crab so they can only be released when you pull it up and open the door. A ring, on the other hand, sits flat on the bottom so crab can come and go, you tie it off on the pier and yank it up fast periodically to check - when it is being pulled up, the crab get stuck in it like a net. Pots cost $50-$100, Rings cost about $30 usually.
Let's talk about bait. Contrary to popular belief, rotting fish guts don't work that well. They want fresh fish pieces. Go to your local market and ask at the fish counter if you can have some fillet scraps. Most stores will give you these for free. If you are an avid crabber, you can get on a program at a fish market where they'll sell you a whole bucket of fish scraps for $5 or $10 on a regular basis. If you can't get ahold of fish scraps, meat works, I've caught them with pork chops before but it's not ideal.
You'll need a license, and a catch record card. When you land crab, you need to record your catch right away. Dungeness need to be male (skinny triangle shape on the underside, females have a fat triangle shape under), and they must be 6 1/4 inches. Spend $1 at an outdoor store to get a crab measuring device. Estimating is a bad idea, tickets for keeping illegal crab range from $119 to $500.
If you don't want to invest in a pot, you can always try your hand at raking. You need an old firm metal rake, not the kind with little tines, but the big thick iron kind. Put on some waders and walk out from the beach into the water at low tide. You'll do best on a beach with lots of eelgrass or similar vegetation. They're not stupid, they hang out where there's some cover to hide in. It's quite the manual dexterity skill to be able to snag a crab with the rake and flip it into your bucket without it getting away, but you'll get the hang of it. Poking the crab to kill it first is extremely ill-advised, as you need to sex and measure a crab to determine if it is a keeper before you end its life, both for good environmental management, and for avoiding tickets.
Most people will tell you that the best crabbing in Puget Sound is in Marine Area 7, which stretches basically from Bellingham to Anacortes and encompasses all of the San Juans.
The best fishing website around if you want to research more in-depth what people are hitting, and where, this year is bloodydecks.com - a word of caution though, fishermen are an ornery bunch, and it can be tricky to get somebody to tell you their secret catch spot.
I've been catching my limit almost every day this year, so I can tell you from experience, it's a good year to be out on the water crabbing. If you have any other questions, let me know, I'll be happy to help you figure it out.
Good luck, and have fun!