It depends. Some schools such as Tuck and Duke give significant importance to networking with alumni. Their expectation is generally that you have done enouygh research about their respective alumni commmunities and you understand what it means to be part of that community.
Others don't care too much e.g. Columbia, LBS, HBS, Wharton come to my mind...
In any case, at the minimum, visiting the campus and networking with alumni add some value to your application when you show that you have invested some time to research the program.
Knowing a member of the board or a donor can be great. But then the level of "intimacy" in the relationship should be quite high to convince them to push your profile. Not many people have that privilege.
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