If the borders of the region or state are closed you will be able to travel there only if you have a residency or a work permit.
Borders can remain open too.
You need to spend resources from the budget to keep your borders closed.
You also need resources to keep your residency issued by the leader.
All this can be regulated by passing the specific laws.
Here you can see the amount of resources needed, if residency is issued by leader and can not be had by citizen, located in the region, without the approval of the leader:
|| Spendings (for 1 minute in 1 region)
You can navigate to Maps and select a filter, which will show where the borders are closed and where they are opened.