You have to keep the wallet open, the computer running and an active internet connection. Please make sure you did put a password on the wallet!
If you can not meet these criteria, you can always export the witnesskey and place it in an Cloud service as offered by https://cloud.g-dash.nl/
The beauty about witness is that the funds will always stay on your account (recoveryphrase) and sending the witnesskey only gives others the possibility to receive the rewards, not your initial funding the witness deposito. GDash will sent rewards back to you every 4 hours (if any received of course). You pay 1 nlg per block you/they witness.