Is it appropriate to ask a prospective PhD advisor whether working for them would be good for my career? Summary Between my Ph.D. offers, I narrowed down my choices to two. Right now, my decision is mostly dependent on a few factors I didn't discuss with either advisor. I want to ask about these issues directly, but am worried about alienating either or both advisors.

My two options

Option 1: 


Advisor is a veteran of the field and some of his recent graduates have landed excellent postdoc positions
Institute is well-established in the field and widely recognized as one of the bests in the world


Option 2: 


Advisor is much younger, though so far very successful. His first advisees will graduate this year (so can't yet gauge their success). 
Institute is very young and its PhD program has had too few alumni so far to give a good image of how successful they are in landing jobs in academia. That being said, institute X has been able to attract excellent faculties and postdocs from top universities (top 10 of the world) and some of their more experienced faculties left tenured positions at those top universities to join X. They also invite world-class faculties and postdocs from the best institutions from all around the world for their weekly colloquium. They're not still as successful in attracting PhD applicants though (perhaps because, as I saw during their interview event, those other offerees that also hold offers from more prestigious universities have the same concern as I do). 
Good program structure and flexibility
Good resources and opportunities 
Tempting financial offer.


My concern: I like option 2, but I want a job in academia. If I choose option 2, the name of the institute I'll have done my PhD at may not carry enough weight to help me get into excellent postdoc positions and eventually land a tenure-track job in a decent university. 

Question: Is it inappropriate to share this concern very frankly with the advisor at institute 2 (who himself has graduated from an incredibly prestigious university)? Could asking that question upset the advisor and negatively influence my chances for later becoming a postdoc in their group? I was thinking of describing the situation as I described above (with a little more details) and ask for Prof. Q's opinion fully honestly. If that's inappropriate, is there a more appropriate way of doing that?
Could asking that question upset the advisor...


You have a legitimate concern. If advisor 2 is angry that you brought up a legitimate, important concern, you're probably better off working with advisor 1. 


  ...and negatively influence my chances for later becoming a postdoc in their group?


It is far too early to start worrying about specific post-doc options.