How to find the largest, most respected web-design companies?

You really ought to figure out what you want to do with your website. Who are you trying to reach? What do you want to motivate them to do when they visit your website? It's almost worth mocking everything up yourself in advance, and *then* finding an agency to do the work - you can see how much it is going to cost. Usually what happens is that a business says "We need a new website" and then figure out some basic abstract requirements. Whoever wins the bid says, "Sure, we can do that" and if you have the money they will do it, no matter how dumb your requirements are. Or, they spend a shit ton of money up front and never nail down the basic functionality you need to engage with your visitors. But if money (and maintaining your sanity) is not a consideration...

I've been working for digital agencies for a large part of my professional career and have worked with some major clients. Your question is a bit complicated. Traditional agencies like WK will do great ad and marketing work but fall short on the digital front, or more likely, a company like Coca Cola will farm their ad dollars to a digital agency. Branding will come from one agency but be implementation by another, etc. I would be very surprised if my agency is not on your list. There's a lot more than implementation: content strategy, social, to things that are more engineering related and would probably be better termed IT consulting with a site tacked on. Feel free to MeMail me and I can give you more detail. I'm a simple engineer but I've seen enough projects to know where good fits are fiscally and culturally.

The biggest, most well-established agencies are going to be, almost without exception, advertising and/or "brand development" agencies. Googling around in those categories should give you a better list to start with. Another good place to look for hits is in a publication like http://www.commarts.com/annuals/2013-interactive. You can get names of agencies from the winners, and look up their web sites to see if they fit your criteria.

Good point on budget Jairus. Going forward let's say no upper limit, just so I don't rule anyone out.

You could try sites like http://agencyspotter.com or http://sortfolio.com.

Any of the super big fully-integrated advertising agencies like http://www.wk.com/ will have a New York office. It's easy to dig up a list of the big guys, but if you're looking for some kind of metric (since your workplace likes going with what's "safe") you could include qualitative details like the biggest clients attached to each agency. Coca-Cola, Nike, Kraft Foods etc. That being said, I'd like to throw one for the little guys - http://www.barrelny.com/ for a smaller boutique agency.

You might find it helpful to check out the list of SODA http://sodaspeaks.com/members/.

On the smaller side, might I recommend http://www.alleyinteractive.com, co-founded by MeFi's own http://www.metafilter.com/user/65126? The firm is small, but it's worked on some really high-profile redesigns, including for The New Republic, Chicago Public Radio, the New York Observer, and Alec Baldwin. I highly recommend their work.

I'm a web project coordinator, I just went through that exact exercise for two different sites. I needed to find one agency that could handle lots of visuals and parallax animation, and one that could do a dense information-driven site. The first thing to do is determine the scope of the project and what range of services you really want. The largest and most-respected firms might not be a good fit for a simple corporate site redesign. Many will offer you a full buffet of strategies and planning and media buys and SEO optimization and social media integration stuff you might not need. I needed design only for my sites, no coding or any extra services as those are handled in-house. Almost all the big firms I contacted insisted on the upsell, so they were right off the short list. Then, think of the kind of site you want and google around to find other sites in the same vein. You'll need to do that anyway to write a decent brief to give the agency. If you find a site you like, find out who designed it. That way you choose your designers based on the strength of their work and fit to your project, not on prestige or slick branding. That's how I ended up finding both agencies I signed. I saw cool sites, called and got the designers referred to me. Ended up with two small local firms, both eager to work with us and willing to tackle only the part of the project for which they were needed. All that at a fraction of the price of the fancy web gurus my boss wanted at first.