Many known networks have structure of affiliation networks, where each of $n$ network's nodes (actors) selects an attribute set from a given collection of $m$ attributes and two nodes (actors) establish adjacency relation whenever they share a common attribute. We study behaviour of the random walk on such networks. For that purpose we use commonly used model of such networks -- random intersection graph. We establish the cover time of the simple random walk on the binomial random intersection graph ${\cal G}(n,m,p)$ at the connectivity threshold and above it. We consider the range of $n,m,p$ where the typical attribute is shared by (stochastically) bounded number of actors.