VNC connection to Mavericks Screen Sharing drops WiFi connection
This happened to me with Yosemite, actually ran into a comment that worked for me here:

http://forums.macrumors.com/threads/mavericks-drops-wifi-when-connecting-in-with-vnc.1694085/

sudo defaults write /Library/Preferences/com.apple.RemoteManagement VNCAlwaysStartOnConsole -bool true

Another option would be to use wired, then it worked for me as well (no setting changes needed).

From https://discussions.apple.com/thread/4170694?start=0&amp;tstart=0 it appears that "maybe" the root of the problem is that it's trying to revert to a login screen (and...doesn't do it right? mine definitely did not ever return to a login screen, it just killed the wifi). So perhaps there's a possibility that if you can get your "wifi" to remain active in login screens (and/or your mac "starts" at the login screen) then you could connect via WiFi and not have it disconnect, and not need the above.  I could not find any way that worked for me to leave my wifi active while logging out, to test this hypothesis, but YMMV.

Another hint: when using wifi with "WPA2 Personal" and the above set to "false" when I connect, it actually works, VNC connects to a "new remote" session, not the currently logged in user (i.e. new login screen) (it also allows me to logout and not disconnect the WiFi, something the "Enterprise" wifi doesn't allow me to do at all).  So my latest hypothesis is that "sharing" stronger encryption connected wifi "WPA2 enterprise" isn't allowed somehow (since it's also disallowed in the "Internet sharing"), and when it tries to open a new session, which is disallowed over the wifi ("sharing wifi" not allowed, for instance, in the sharing pane) it forces a disconnect.  But connecting to the current user is OK.