Three things, let's suppose that your AIX system has no special protections.
1) Most of Microsoft Windows systems have default settings saying that the telnet session must be closed after certain time without activity. So, your PC by itself will close the putty telnet session. Look on your Microsoft Windows network settings to see if it's easy to change that setting.
2) If your global site experiences a network failure (cable removed or anything else) you can loose your telnet session. Then, if you are running a scripts which does not need input from your hands, you may consider starting this script with the nohup option. for instance try "nohup sleep 10 &". This will put the session output in the file named nonup.out, and will survive any loss of telnet connection.
3) Personally, I use vnc emulation. I start a session on the AIX system, and then I connect with vnc client.
If you startup something inside a vnc client session, you can reconnect to this session from another terminal, or from your home ! And if your pc is powered off when you come the next morning, you power on again, enter inside your vnc session, and have your screen alive and you can continue your backup process.