Why did austrians want to keep Louis XVI as king?

There were two main reasons:

(1) Marie Antoinette, King Louis XVI's wife, was from the Austrian noble family. If the French monarchy would be overthrown, there was a serious risk for Marie Antoinette's safety. These concerns were eventually proven when the French executed her.

(2) A threat against monarchy anywhere was a threat against monarchy everywhere. The arguments of the French Revolution could be turned against the Austrian monarchy.