Can the mobs climb ladders?

Solution 1:

While mobs are technically capable of climbing ladders, the pathing AI generally prevents them from doing so - instead, they will typically just spin around in circles.

In general, the best way to restrict access to a shelter is to build a Door.

Solution 2:

I have personal confirmation that if a mob is chasing you (in my case it was a skeleton) and you go up a ladder, it will follow you up it if it can.

^{And it's absolutely terrifying.}

Solution 3:

As an aside - if you light an area with torches it will prevent zombies etc. spawning there. So add some torches in your lower area for a zombie free experience.

It doesn't stop them wandering in from somewhere else, so you'll still need to build walls, doors etc. to prevent entry.