When trying to establish a connection with a STEM field professor, is it appropriate to ask them out for coffee? I have a friend who is trying hard to establish a connection with a well-known professor at a generally top 5 school for STEM in the U.S. This professor is frequently at some good seminars around the area.  

My question is: is it appropriate for my friend to ask out this professor for coffee?  She has tried several times - trying to meet him one on one before the start of seminars, some other scheduled times, etc.  This may be relevant, so I will note that she is female, and the professor is male.  

He has turned my friend down with reasons such as "I'll be traveling and can't meet for coffee", or "sorry, but I won't be able to make it today." 
Otherwise, the professor answers all of her questions and gives great advice to her, in terms of how to advance in the field.  Based on what she tells me, he is very polite and helpful.

Personally, I would never ask a professor out for coffee - let alone try repeatedly, after he or she has said "no" several times.  I would email to start some dialogue and perhaps meet in his / her office.  So I just want to know whether perhaps I should hint to my friend to stop trying to get him out to coffee and that maybe he's not so into the idea of letting students schmooze with him one on one.

I wonder if this is sort of specific to the STEM field academia culture, too; e.g.  if we were in film school or business school, I think it'd be much more "normal" to ask a professor out to coffee to try and build a relationship.
I don't think that this is special to any type of professional environment. The professor is polite and responsive. However, they repeatedly tactfully refused an informal meeting, which indicates that they are uncomfortable with it (indeed, they may be coincidentally busy at the given time, but often this would be reflected in the suggestion for rescheduling, which I don't see). In my opinion, your friend should take the hint and reevaluate their approach, e.g. schedule an office meeting.

As to whether it is appropriate to ask, as long as the tone is professional, I see no problem in asking, just as I see no problem getting 'no' for an answer. This, however, may in some cases be culture specific.