When you talk about smuggling, whether or not border is closed, smuggling takes place. So, rather than a government trying to close bother totally, they should instead try to reinforce the agencies in charge of checking the Inter country movement of goods and people.
If you because of bad eggs and shut the whole border against a country, it will bounce back to your people when your business will begin to cripple.
The world is connected, everyone country needs another's help to survive.