Note: this answer is not as applicable for cultures with higher respect for authority, but more Western cultures (where hierarchy is less important to respect).
Is it a good idea? Or crazy?
I don't think it's difficult to network with higher level folks as an intern. In fact, I think that's perhaps the best time to do so. Everyone expects you are looking for career guidance/direction and it's 100% ok to do so overtly.
When I was an intern, I actually did exactly this. I sent an email to the VP of the division I was working in - for a 50k+ company - and got together with him for some time to talk about career perspective, paths, etc. It was great, totally intimidating, but really worthwhile. Another intern went to the same school as our CEO and got together with him (all the other interns thought he was CRAZY for asking about that).
Something to think about - what percentage of interns even try to do this? 1% maybe? Most non-interns won't, either. If all interns were doing this it might be harder but most are terrified of the idea.
High level executives won't be someone to personally refer you, etc. But they can provide you great networking contacts and wisdom.
How to approach it
I imagine a conversation along the lines of:
... this isn't how most conversations like this go.
A lot of folks in higher level management positions enjoy mentoring/coaching people in career perspective. People generally like to help others, too. My experience has been that people in higher level management positions are more willing to have these types of conversations, because even though they are super busy, they get nearly no requests for this and often like doing this sort of thing (connecting people, giving career guidance, etc).
People love to give advice, especially when they think it will be useful.
So what you need to do is make it clear that:
- You are an intern/junior employee
- The VP has something you would benefit from (experience, wisdom, etc)
- You would love to get their perspective and some of their time
If you do reach out, email is a good way to do it, something like:
Hey Mr. VP,
I am an intern here and would love to get perspective from you on my career. Would you be willing to get together? If so, I can setup some time on your calendar.
But be very sure:
- Come with a specific list of questions to ask. You don't need to do go through them (VPs often like to talk and can be good at guiding this convo, but make sure you are prepared)
- DO NOT WASTE THEIR TIME. This should be obvious, but if you are getting time with a VP don't waste their time.
- Send a followup, "thanks for being willing to get together - I appreciated your guidance and it will be very helpful as I figure out my career!" type of email. If you get people to connect with, consider sending another followup email a few weeks later like, "I got together with X and it was really beneficial - thanks again for the advice!" But make sure these emails are 1) very easy to read and 2) very obviously not a "needs response" type of email.