Either way will work but if you can I would just leave the client running. You can't control other nodes all you can do is "talk to them" through your client (happens automatically). Right now the majority of the network believes your old tx is valid and will reject the new one. 37 nodes believe the new tx is valid and will reject the old one. So the network is "split" eventually one of those tx will end up in a block. Since other nodes know about your tx and will continue to try and relay it you probably can close your client but having your client broadcasting is one more node "on your side".
IF all that was excessive. Simple version:
If you can leave the client running, do so as it can help spread the new tx faster.
If you can't leave the client running you should be fine as 37 peers know of the new tx however if they all went offline your tx might not confirm until you start your client up again.
And up to 67 nodes now.