On Ronald Brown's, pg281, 7.3.8 His proof is summarized as
If $(w,h), (u,k)$ represent $(X,A)$ and $(Y,B)$ as Strom structures, then $(l,v)$ represents $(X \times Y, X \times B \cup A \times Y)$ as a Strom structure, where $v(x,y) = \min (w(x), u(y)).$ and $$l=(h(x, \min(u(y),t)), k(y,\min(w(x),t))$$
Suppose $t>v(x,y)$. I need to show $l(x,y,t)$ is in the union $ X \times B \cup A \times Y$. But this is not clear(?)
If we consider the special case $t> w(x),u(y)$, then $l(x,y,t)= (h(x,u(y)),k(y,w(x))$. But we cannot tell where this lies if $w(x)=u(y)$.
Don't hesitate to consult the original papers by Strom referenced in the book.
However, the missing argument is this:
We know $h(\lbrace x \rbrace \times (w(x),1]) \subset A$. Because $h$ is continuous and $A$ is closed we get
$$h(\lbrace x \rbrace \times [w(x),1]) \subset h(\overline{\lbrace x \rbrace \times (w(x),1]}) \subset \overline{h(\lbrace x \rbrace \times (w(x),1])} \subset \overline{A} = A .$$
In particular $h(x,w(x)) \in A$. Here you have a reason why closed cofibrations are particularly nice.
By the way, one can show that 7.3.8 remains true if we require only one of $(X,A)$ and $(Y,B)$ the be closed (but of course we need that both are cofibred pairs).