Even Reuters' international correspondents are unclear on the concept of borders and border security, offering this:

Iraq's government said on Tuesday it would close its borders with Syria and Iran and extend the hours of a night curfew in Baghdad under a U.S.- backed security plan to rein in violence in the capital.

The measures ordered by Shi'ite Prime Minister Nuri al- Maliki were announced on Iraqiya state television by Lieutenant General Abboud Qanbar, the Iraqi commander who is leading the U.S.-backed security offensive in Baghdad.

Qanbar said the border with Iran and Syria would be closed for 72 hours...

The article is prepended with this:

Corrects to show government says will close the borders, not that it has already shut them

I assume that this means they will be closing border crossings. What actions they take to prevent illegal crossings aren't known, but I'm sure they aren't as stringent as they could be.

If I recall correctly, shortly after the invasion it was pointed out that we didn't have enough troops to prevent illegal crossings and that this was a huge security risk. That was contemporaneous with the rise of foreign fighters entering the country and an insurgency forming. And, I believe that such concerns were brushed aside by the Bush administration as well as the usual hacks outside.

I'm also sure it wouldn't be difficult to find people arguing at the time that securing the borders would be a great make-work project for newly unemployed Iraqis, and, while that would have led to corruption and infiltration of that force by insurgent supporters, it also would have greatly reduced the numbers of people available to join the insurgency and given them something to do and some income rather than just throwing them out on the street. Needless to say, such intelligent plans didn't find favor with the Bush administration.