You're using Putty.  All you have to do is start your remote miners with the nohup command added to them.  Example:

nohup ./bfgminer -o stratum+tcp://stratum.somepool.com:3333 -u workerName -p workerPassword &

If you were to use screen as was suggested earlier, you'd have to install it on the remote machine.  Not sure what Linux flavor those remote machines are running, but if they're on a Debian-derived distro, you'd just do this:

sudo apt-get install screen

Then, you'd start your miners like this:

sudo screen -dmS MyMiners ./bfgminer -o stratum+tcp://stratum.somepool.com:3333 -u workerName -p workerPassword

This starts up a session in the background.  To access it, you'd just do this:

sudo screen -dR MyMiners

Then, once you're in the screen, to get out of it, all you need to do is hold down the control key and hit a then d.