More, or more often? (Or both?)
A bit of both, truth be told - those some slower bowlers don't swing it all that much (i.e - Stuart Clark) whereas some faster ones do (i.e - Fleming).
Having an action conducive to swinging the ball helps (an open chest apparently results in less swing), of course. But where two bowlers have perfect actions (an impossibility, but bear with me

), I'm inclined to believe that the slower bowler will swing it more.
Either could, conceivably be true - certainly most agree that there's an "optimum pace" for achieving maximum swing in a given delivery, but a) it's still possible to swing it plenty at much higher (or lower) pace than that optimum and b) I've never yet seen any suggestion of a bowler's high pace handicapping his ability to swing a ball - ie, a bowler sending down most of his deliveries at ~125kph managing to swing a ball that a bowler sending down most of his deliveries at ~145kph is finding impossible to.
Well again, I'm not trying to argue that express quicks cannot swing the ball - merely that a slower bowler with an identical action will swing it more (maybe because the ball spends more time in the air).
It certainly can hinder you, for all sorts of reasons (sometimes when you, naturally, do everything perfectly then trying harder will disrupt the equilibrium - Massie himself would be a good example of that), but there's no doubt you've got a better chance of achieving swing if you hold the ball seam-up rather than cross-seam - ie, if you have the aim of getting swing rather than if you have no aim of getting swing. And I don't just mean trying on a delivery-by-delivery basis either, I mean putting in the effort in the nets and honing your technique so that you use the seam as best as possible.
Well most certainly, action and seam position are important when determining how much you'll swing the ball - but Massie's point was that having all of that (as he most obviously did) and bowling too fast ("putting too much into it") will result in less swing than otherwise. There will still be some swing, but not as much as you'd probably like.
Regardless of his pace, a bowler who works on being a high-class swing bowler will be a better one than one who doesn't. Swinging a cricker ball is one of those things that can be almost entirely taught\learnt, if you've got the tutor\work-ethic.
Mitchell Johnson seems to disagree. He goes from perfect seam to cross seam on a whim, which is annoying. In other words, he seems to learn it and has the requisite work ethic, but he cannot consistently apply his knowledge.
Humidity and cloud certainly helps, but top-quality swing bowlers with a good-quality ball can achieve clear-air turbulance on the driest, sunniest days. There are almost no conditions that can stop the ball swinging - though a bad cricket-ball certainly can. But there are plenty of conditions that can make it difficult\impossible to seam\cut\turn the ball.
As Chris Old et. al will tell you, swinging the ball is rather difficult on dry, sunny days - and he was good at moving the ball! I admit that I don't see how that's significantly different to, say, finding it hard to cut/seam the ball in some conditions.
EDIT: Actually, Chris Old was adept at using both seam and swing. Heh.