My understanding is that those stabbed by a Morgul blade slowly turn into a wraith.
Is there a reason the Nazgûl and the Witch King of Angmar did not go around stabbing people and turning them into wraiths?
I'm reasonably sure the only person other than Frodo who Tolkien mentions being stabbed with a Morgul-blade is Boromir, a Steward of Gondor. The section of Appendix A to Lord of the Rings about the Stewards says:
In the last years of Denethor I the race of uruks, black orcs of great strength, first appeared out of Mordor, and in 2475 they swept across Ithilien and took Osgiliath. Boromir son of Denethor (after whom Boromir of the Nine Walkers was later named) defeated them and regained Ithilien; but Osgiliath was finally ruined, and its great stone-bridge was broken. No people dwelt there afterwards. Boromir was a great captain, and even the Witch-king feared him. He was noble and fair of face, a man strong in body and will, but he received a Morgul-wound in that war which shortened his days, and he became shrunken with pain and died twelve years after his father.
That is, not everyone stabbed with a Morgul-blade becomes a wraith; some of them just end up with painful and eventually-deadly wounds. Since becoming a wraith is otherwise correlated with Rings of Power, it seems like a reasonable guess (though not one otherwise supported, to the best of my knowledge) that Frodo would only have become a wraith because he was stabbed while being a Ringbearer, as Jardyn suggests in the other answer.
Aside from the other answers, it's wrong to think of a Nazgul as some kind of Mr Stabby Mc Stabstab that likes to run around Middle Earth creating little wraiths.
Nazgul are sinister, not psychopaths. They're a tool, a weapon that is only used at the greatest need. Witness Radagast's conversation with Gandalf in FotR:
'"I have an urgent errand," he said. "My news is evil." Then he looked about him, as if the hedges might have ears. "Nazgul," he whispered. "The Nine are abroad again. They have crossed the River secretly and are moving westward. They have taken the guise of riders in black."
'I knew then what I had dreaded without knowing it.
'"The enemy must have some great need or purpose," said Radagast; "but what it is that makes him look to these distant and desolate parts, I cannot guess."
"The Nine are abroad again" - "The enemy must have some great need or purpose" - these are not beings that run around willy-nilly, as I said.
"Purpose" is a key word here, we need to look at what the Witch King's purpose was in stabbing Frodo. Was it to make him a wraith? Certainly not. His purpose was to get the Ring back for Sauron; making Frodo a wraith was a means to an end, not an end in itself.
We learn a good deal more in The Hunt for the Ring, published in Unfinished Tales. First of all:
At length he resolved that no others would serve him in this case but his mightiest servants, the Ringwraiths...
This backs up my first statement - the Ringwraiths are not a tool that Sauron uses lightly; they are his mightiest servants that he uses only for the gravest tasks.
Yet this weakness they had for Sauron's present purpose: so great was the terror that went with them (even invisible and unclad) that their coming forth might soon be perceived and their mission be guessed by the Wise.
So, if the Ringwraiths did run around Middle Earth creating little wraiths, their presence would very soon become known. That would hugely impair their ability to act in secret:
his (Sauron's) chief purpose was that the coming forth of the Nazgûl should appear only as part of his policy of war against Gondor... the Nazgûl were commanded to act as secretly as they could.
It's clear that running around Middle Earth creating little wraiths would be quite contrary to Sauron's purpose.
So in conclusion we have two reasons against running around creating lots of little wraiths: there's no additional purpose to doing so, and doing so would risk comprimising the Nazgul's secrecy, and thereby go against Sauron's plans.
As I understood it the reason that Frodo was in danger was because a fragment of the morgul blade snapped off and remained in his body, working its way towards his heart. In the books it is mentioned that at that point he would have become a wraith and was only saved because Elrond removed the fragment. This would not necessarily happen in all cases though, so I have never inferred that simply being stabbed by such a blade could cause this to happen - only if it fragmented in the wound. As Gandalf explains to Frodo in Rivendell:
They tried to pierce your heart with a Morgul-knife which remains in the wound. If they had succeeded you would have become like they are, only weaker and under their command
My understanding of it is that if the person stabbed does not have a ring of power, he will die, however if he/she does, then the blade has a sort of corrupting and enhancing power on the ring.
Sauron gave 9 rings to 9 powerful leaders of Men. These 9 all became Ring Wraiths upon their deaths because of the Rings. Frodo being the Ringbearer, if he would have died while holding the ring would have become a Wraith as well. Further if Frodo would have died due to to wound inflicted by the Nazgul, he would have been subservient to the Nazgul as a Wraith.
My understanding is that there were wraiths... and then there were wraiths. The Nazgul were powerful and feared as wraiths because of the rings they possessed. But a typical person who became a wraith would exist only as spirit, with little ability to influence the physical world.
Note that fragment of the Morgul-blade snapped off and remained in his body. The remaining part of blade "seemed to melt and vanished like a smoke in the air".
This weapon almost certainly may be used only once and probably is troublesome in storage (destroyed by light) and hard to produce.
Therefore producing new wraiths, probably far less powerful than Nazgûl using this method is inefficient use of available resources (both Ringwraiths and materials necessary to produce blades).