Oh so it must be always running? I didn't realize that. Also does the wallet have to be encrypted?
To generate POS blocks you have to be connected to the network, so yes, the wallet needs to be running for any reward generation to happen. Remember, these blocks are what is keeping the network secure and verifying transactions.
Encryption is simply for wallet security. Unlocking is what is needed for your coins to enter maturity and be ready for staking.