More than 100 migrants, the majority of whom claimed to be Iranian, have crossed the Channel from northern France this month.
French and British officials will work together to maintain border security from the base in Calais.
The centre began operating on 23 November.
Lucy Moreton, general secretary of Immigration Services Union, said attempts to cross the Channel had placed Border Force officers under “appalling” pressure.
She said: “This is a dangerous crossing. Sooner or later there is going to be bodies in boats. There aren’t yet. It hasn’t happened, or not that we know of, but it will eventually.”
The BBC has confirmed that at least 110 migrants have entered British waters since 3 November. But Ms Moreton told BBC Radio Kent the total number of migrants rescued by British authorities was “probably” closer to 130 or 140.
The Coordination and Information Centre, which is part of an agreement between the two countries, will see Border Force officers working with their counterparts in Police Aux Frontières.
The UK has spent £44.5m on border security as part of the Sandhurst Treaty.
At Prime Minister’s Questions, Charlie Elphicke, (independent) the MP for Dover and Deal, asked Mrs May about the “unprecedented” number of migrants who have crossed the Channel in “unseaworthy craft”.
The PM said: “In our discussions with the French government, we agreed that we could set up a co-ordination centre which would enable the French and UK governments and authorities to work together on exactly these sorts of issue.”
The centre will focus on intelligence sharing to combat cross-border crimes.
The Home Office said: “Importantly, it will also help with the identification, prosecution and dismantling of the organised crime groups currently behind illegal migration attempts by small boats across the Channel.”