There is a bug with Cisco AnyConnect VPN client and OS X Mavericks that will cause the VPN connection to randomly disconnect. It happens using Wi-Fi or Ethernet.
Below is a bug description and they suggest a possible fix but I can tell you that I’m using a later version of Cisco AnyConnect than the one mentioned below and then issue is still there. As of this writing, the latest version is version 3.1.05182.
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Description
Symptom:
AnyConnect VPN connection is frequently interrupted on OS X 10.9 (Mavericks) due to bug in Mac OS X 10.9 (Mavericks) - Reference Apple bugreport ID: 15261749.
Conditions:
After connecting, users will experience frequent interruptions to the VPN connection
Please note that the reliability problems do appear to vary based on WiFi/Wired network connections, although both are impacted.
Workaround:
We are pleased to announce that we have made available a temporary workaround (CSCuh73010) for the AnyConnect Mac OS X 10.9 connection instability / compatibility issue (Apple Bug Report ID 15261749) via an updated version of AnyConnect (3.1.04074). There are caveats associated with this temporary workaround which are noted in the release notes.
http://software.cisco.com/download/release.html?mdfid=283000185&flowid&softwareid=282364313&release=3.1.04072
(It is located in the 4072 directory.)
Release Notes: http://www.cisco.com/en/US/products/ps10884/prod_release_notes_list.html
Note: A SMARTnet-enabled login is required to download AnyConnect Software from Cisco.com.
Here's the release note excerpt: This update is a temporary workaround to resolve the problem of frequent reconnects/disconnects of the AnyConnect VPN on systems running Mac OS X 10.9. Apple is aware of this issue and you can reference Apple Bug Report ID 15261749 if you want to open your own case with them. This patch is a workaround and we are not able to guarantee that it will work as reliably as a fix provided by a future OS patch. A future OS fix will not require you to update AnyConnect 3.1.0472 to operate reliably. Once Apple provides a fix in OS X 10.9, we may choose to retract this workaround, and unpatched versions of AnyConnect 3.1.04072 will also work reliably.