Much as I'm opposed to gay "marriage," I've never seen this claim adequately defended (and it's usually not defended at all, but simply asserted).
Well, it will destroy the institution. First, it will re-define it. That fact is obvious. However, marriage can not continue to be re-defined and have the people expect that definition to have any meaning. Note that marriage has been under rather continual re-definition since sometime in the 1960s. Eventually, the word "marriage" will mean whatever the people using the term want it to mean. At that point, the institution is dead.
I submit that this point is when two people of the same sex can "marry." The word will be meaningless and the institution will be dead.
Shalom.
So long as men and women continue to enter into real marriages, I don't see how the institution is destroyed. As Taliesan posted, "Calling a donkey a horse doesn't hurt the horses."
Marriage has been under rather continual re-definition throughout history. This is true in both U.S. (see, e.g., Significant Changes in U.S. Civil Marriage) and Western history.