Normally there is no passport check.But it still is a border and sometimes there are surprise checks, so bring your passport or European national ID.By the way, you will not travel on an Eurostar train but on a TGV or a Thalys train. The same kind of train but a different colour.Or you can choose to travel on the &m76;nor2al8#8211; trains, the ones for which you do not need a reservation, but those mostly use a different station in Lille.References : I have traveled that route several times.