With very few exceptions, puppeteers of hand/rod puppets like the Muppets put their dominant hand inside the puppet's head. There are thousands of photos that demonstrate Henson used his right hand for his characters' heads, like this one:
Or this one:
But hey, what about this one:
He's using his left hand there, yes? Actually, no. As could very easily happen back in the old days of film and negatives, this photo has been mirror-flipped. You can tell by the way his shirt buttons, and where the pocket is, and the direction of his hair part. It should really look like this:
Okay, so Henson puppeteered right-handed. But what about writing and drawing? Well, here's a clip from the 1968 film "Muppets on Puppets", where Henson draws with his right hand as he talks to puppet builder Don Sahlin:
Here's a one-minute clip from 1994's "The World of Jim Henson" with ten shots that show him holding and using a pencil with his right hand:
'Nuff said. So, how did this Henson-was-a-lefty rumor get started? Perhaps it is based on the fact that most Muppet characters seem to be left-handed. This is most obvious in a group like the Electric Mayhem, where characters play guitar "backwards", like left-hander Paul McCartney does:
But this is only a function of the puppeteers themselves being right-handed. The puppeteer's right hand goes inside the puppet's head, leaving the puppeteer's left hand free for simulating the guitar playing. If you're going to fake guitar playing by moving only one hand, the "strumming hand" is a much more effective choice than the "chording hand". And while the puppeteer could reach across their body and use their left hand to manipulate the puppet's right hand, if you've ever tried it, it feels very strange. It's much more natural to just flip the guitar around, and use the puppet's left hand for strumming.
Hence, most Muppet characters seem to be left-handed, and I suppose this has led people to assume this means the puppeteers are too. While there are left-handed puppeteers who have worked with the Muppets (like me), just like the general population, Muppet performers are largely right-handed. And so was Jim Henson.