I believe I have answered my own question. And hopefully this can help anyone else with the same sort of need. The answer came in my comment that the machine used to access the remote server was only monitoring and there was no direct interaction. Well then the computer becomes somewhat inactive even though we have set it to not go to screen saver. So I started process that keeps the machine busy. Since this we have not lost connectivity. However we did find some stability on a Windows 2000 machine which I quickly updated to XP. Problem appears to be taken care of now. Thanks again to all for the good feedback.
Stay happy and safe......