I was in early beta and the Wailing Prison starter dungeon could be skipped with new characters as long as you played through it once. It appears that that is not the cast now. Unless I'm missing something?
You need to play it through once (and beta doesn't count... you need to play through it on the live game). Then, on your second character onward, it will ask if you want to skip.
I recommend not, though. The loot drops, particularly weapons and armor, can make your early game experience much less painful.
Living vicariously through my characters.
My Girls:
"If you were trapped in your house for, say, a year, how would you pass the time?"
Nephikah the Houseless, dunmer assassin: "I suppose I could use the break. I have a lot of business holdings now that need management." Swum-Many-Waters, elderly argonian healer: "I think that I would enjoy writing a memoir." Silh'ki, khajiit warrior-chef: "Would this one be able to go outside, to the nearby river? It's hard to fish without water!" Peregrine Huntress, bosmer hunter: "Who is forcing me to stay inside, and where can I find them?" Lorenyawe, altmer mechanist: "And why would I want to go outside in the first place? Too much to be done in the workshop." Lorelai Magpie, breton master thief: "I'd go nuts. Lucky for me, I have a little experience sneaking out!" Rasheda the Burning Heart, redguard knight: "I would continue my training to keep my skills sharp." Hex-Eye Azabi, khajiit daedric priestess: "I suppose it would be lucky, then, that I built a shrine to Mephala in my backyard." Yngva Stormhammer, nord bandit (reformed...ish): "I hate being inside even when I'm not forced to be. GET. ME. OUT." Madam Argentia, vampire dunmer aristocrat: "I suppose it would be more of the same. I have a rather... contentious relationship with the sun." Mazie gra-Bolga, orc scout: "Uh... I'd have to house train my bear..." Felicia the Wanderer, imperial witch-for-hire: "What Lorelai said." Calico Jaka-dra, retired khajiit pirate: "This one would like a rest from her grand adventures. Her jewel shop runs out of stock!" Shimmerbeam, blind altmer psijic: "Provided that I am confined to Artaeum, I do not think I will want for things to occupy my time." Shauna Blackfire, redguard necromancer: "Sounds like paradise. I hate people." Kirniel the Undying, cursed bosmer warrior: "I would feel useless, not being able to fight." Echoes-from-Dragons, argonian who thinks she's a dragon: "All the better to count my hoard!"
This doesn't appear to be the case any longer. I just played through with a newbie character, then started another one. Put me right back in the Wailing Prison.