They went straight through customs with nothing to declare. 

=They went straight through customs because they had nothign to declare.

赞同蒋老师解答观点，我补充一点，判断with +名词+to do 是否是复合结构，方法是后面的名词必须是不定式逻辑主语。显然网友给出句子with nothing to declare中，nothing不是不定式to declare逻辑主语，不定式to declare逻辑主语是句子主语they.(相当于with nothing to declare=because they had nothing to declare) 所以 with nothing to declare不是with复合结构，而是with+名词+后置定语（to declare) 构成介词短语作原因状语。