For now, there is no possibility to enable/disable a Composer environment. In order to save money on a server that is not in use, there is a need for a feature similar to autoscaling, for which a request has already been filed.
On Medium site, you can find a lot of useful information, regarding saving costs.
One way to control your costs in Cloud Composer is to use autoscaling. The amount of nodes can be set to autoscale in GKE cluster, follow this guide. Smaller size of Cloud Composer environment and shorter running time would be best practice.
Cloud Composer charges for compute resources allocated to an environment, components continue to run even when none DAGs deployed. There's not much you can reduce/turn off, you may consider another platform services, such as Dataflow, which is serverless.
I hope you find the above pieces of information useful.