While the management of allergens can be difficult for packaging, I believe the safest way around it is to be proactive and not reactive.
Depending on your local regulations pick out the top allergens, for us here in the US those would include:Milk, eggs, fish, crustacean shellfish, tree nuts, peanut, wheat, soy.
You could begin by doing surface testing of each one of these allergens and qualify them by an ELISA method to determine which one of these would pose your highest risk and begin a monitoring program. Keep in mind that the surfaces that you pick have to be relevant to the amount of activity and the risk that it poses to the packaging. Once you have validated your method you can do random spot checks with a quick lateral flow device of the most prevalent allergen, continue to qualify with ELISA annually from there on out.
As far as control goes, you can have limitations in what food goods are allowed to be brought in to break areas and a zero tolerance policy for food or drinks on production floors. An additional layer of control can include letters of guarantee from raw material suppliers and plant services guaranteeing that none of the products that are being brought into your facility contain no allergenic material.
Train all employees to understand the importance of your rules and follow your GMPs!