Befriending someone for their connections is networking with a disguise on.

Wanting to meet useful people is not the problem; the trouble starts when the wanting dresses itself as affection. The other person usually senses the shape of it before they can name it, and what stays with them afterwards is the pretending rather than the ambition. Honest interest in what someone does is sturdier than manufactured warmth, and it survives past the moment they can help you.

Take a relationship you have been cultivating and ask what you would do if they could offer you nothing. If the answer is not much, say the real thing instead. Ask plainly for the introduction and let the friendship stand on its own.