Which band was the most successful?

Nope. As someone who studies album sales of certain bands intently I have to say I disagree with some of this.

First of all it's true that the first three albums went gold but in the case of the second and third only a few years later and not when they were first released. The first album didn't go gold until the 90's. Also, POKR and Leftoverture went four times platinum not more than that. If they sold more it hasn't been documented by the riaa(maybe someday but not as of yet). Finally, Monolith did not go platinum. It "only" went gold. If it has gone platinum then it hasn't been documented. It is certainly possible albums have sold more than what the riaa reports(they can only certify something if it's requested and since it costs money to make that request not all record companies do that until they are sure it's worth it if they do it at all)but we can only know what is official. 

Artistic success = King Crimson

Influential Success = Really only one of those bands falls away and that's ELP. All the others have been largely influential.

Well, like I said it's quite likely they have sold more then what is listed anyway. That holds true with other bands. For example, Dark side of the moon is listed as 15 million but that is from 1998. I'm pretty sure it has sold at least 22 million by now. Close to the Edge is certified as platinum(also 1998) but has probably sold at least 1.5 by now and maybe even 2 million. Who knows. There's probably other examples too. 

Monolith was listed on wikipedia at one point as selling one million but that hasn't been verified on the riaa website so it can't be considered official. I'm an editor on wikipedia so I had to change that so it only showed gold status. Trust, me that was hard to do. ;) I dont mind someone changing it back as long as it reflects the truth but so far no evidence to suggest it has gone platinum yet(only gold).

I'd like the OP to make it BOLD it's not about commercial success.... or even rethink the title, because frankly, successful = wealth + fame + recognition, mostly!

Back to the topic.

Pink Floyd - inspired many people outside of prog, but their most "famous" albums are easy listening really

Genesis - too little recognition in the 70s, maybe the most "obscure" of the lot in pre-Duke era. It must be said they influenced a heck load of bands later on, especially in the Neo Prog days

Yes - well they had recognition in prog days, had the money, inspired some artists - although their style and technical prowess was hard to mimic. Definitely a very successful and quintessential band, could be my pick

Rush - not that proggy, I think they influenced the nerdy crowd and maybe 90s prog metal, but for prog rock, they arrived too late

ELP - maybe the biggest name in prog rock, commercially, in the 1970-73 era. Emerson was a household name, Lake was too, and they lived up to the hype. They really fell out of favor later and never really "redeemed" themselves outside of prog circles. As richardh mentioned, "self parody" was a problem

So I pick King Crimson

There's alternative and metal fans who like KC. The same thing with Porcupine Tree. I would say all of these bands probably have some kind of following or fans outside of strictly prog circles though. That's what happens when a band reaches a certain level of fame and notoriety. With some it's probably more on the classic rock side(PF, ELP, Yes) and with others it's more indie and alternative(KC and PF and to some degree Rush). Genesis probably are a mixture of those with probably a lot of "pop" fans also. Most metal fans are probably more into Rush, KC and PF then any of the others though.

Most people who like prog anyway.  

and the ranking was:

KC

Floyd

Yes

Genesis

ELP