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In Game of Thrones S08E02 "A Knight of the Seven Kingdoms", it is repeatedly mentioned by multiple characters (I counted six) that when the Night King and the Army of the Dead attack, the Winterfell crypts are the safest place to be for women, children and old people.

But why is that? As far as I can see, it's got a single way down with lots of steps and poorly lit. If the dead win, the people there will be slaughtered with no way out. In any other place, they would have a chance to run at least.

Why would this be the safest place?

It is the safest place while the battle rages on, similar to the Battle of Helm's Deep in LOTR: while in the crypt/cave, the non-combatant are out of harms way, and in no risk of being hit by a stray projectile or shrapnel. Plus, if the defenses are breached, then there is no risk of any number of civilians falling in the hands of the enemy to be slaughtered and join the Army of the Dead.

Of course, if they lose the battle, then everyone dies, but that's why they are fighting in the first place: to avoid being wiped out.

If they win, then the civilians, used in the modern sense opposed to military people, are safe and sound and can tend to the wounded combatants who survived the battle and start rebuilding and such.

As for running, it's highly unlikely that the women, children and old people would go very far before being caught by the Army of the Dead and slaughtered in the open fields. Especially since winter has finally come. There is no running from the Army of the Dead. Besides, from a strategic standpoint, any way out is also a potential way in.

Sava's answer is correct, but I wanted to respond to part of your question.

But why is that? As far as I can see, it's got a single way down...

You see it as "only one way out". However, someone who is defending sees it as "only one way in", which means that when you are able to defend this chokepoint, you can keep everyone safe.

This is exactly why castles have a wall and a handful of gates (or often just one): the less gates into the castle, the more resources/manpower you can dedicate towards defending these gates.

To paraphrase The Man In The Iron Mask, when the commander explains to the king why they cannot beat the four musketeers who are entreched in a hallway: "The chokepoint of the corridor negates our superior numbers".

with lots of steps

The civilians won't be moving much when they're in there, why would the steps matter?

& poorly lit.

Again, there's little reason for the civilians to need good lighting while they're hiding. They're not doing anything. I also don't see why they couldn't just light the crypts better if they really needed to.

Additionally, for a human enemy, should the enemy enter the crypts, it makes you more likely not get spotted when you hide. Or even when you do fight back, your eyes are used to the dark and the enemy's are not yet, which again gives the defender the advantage.

Why would this be the safest place?

Because statistically it has been. Consider that White Walkers haven't been seen for a thousand years, but human wars have happened pretty much every generation. This is a centuries' old habit: hide the civilians in the crypts, because it's the best place to keep them safe from the battle that rages on outside.

There is a strong consensus on the tactical benefit of hiding the non-combatants and to keep them safe. This is for two reasons: the soldiers don't need to worry about the safety of they family and can instead focus on the battle, and keeping the civilians safe means they can assist with the post-battle fallout (tend to the wounded, clear rubble, restore things).

They did it in King's Landing with the Battle of Blackwater Bay (it's the scene where Cersei talks to Sansa about having instructed the guard to kill her and her children should the safe place be breached, so that she is not taken by the enemy). Jumping to another franchise, one of the best known use cases of this tactic is in Lord of the Rings, in the Battle for Helm's Deep.

You should consider this safe place as a "keep within the keep". They are the second walls, when the first wall is taken by the enemy.

This is the best way to assure their safety and have the most probable outcome for winning. If the non-combatants were to be hidden away in a different town farther inland, there is still the chance that a part of the undead force could break off and attack (or even the golden company). This would mean that the troops at Winterfell would need to detach a size-able group to accompany them to protect them "just in case". They are considerably outnumbered now and need every fighter they can get, so splitting the forces wouldn't really be a feasible option.

With the non-combatants staying IN Winterfell, not only are all of the troops together in one place to provide the best offense they can, they are also assured and not as worried about their loved-ones well being. They know where the non-combatants are, underground (out of the reach of the enemy), and the only way they will be come endangered is if all the fighters at Winterfell are defeated, in which case, they would be doomed no matter where they were.

With their loved ones near and depending on them, the average person is also probably willing to fight harder to make sure they stay protected. This is the last stand, they're putting all of their chips into one basket because to split them up against a force as large as the undead would be result in decimation.

There is a slightly more morbid reason to have the non-fighters corralled in a single place. It's so that, if the defenders can see they're losing the fight, they can themselves kill and torch them humanely before killing themselves, so prevent them from being captured and turned. It's reasonable to expect the people hiding to panic and try to run in such a situation, but from humanity's point of view it's more important that the enemy's army doesn't grow more than it has to.

Those stone tombs are too heavy for the dead to open. So even if the Night's King raises e.g. Ned and Lyanna, they wouldn't be able to get out of their crypts to attack the living people.