Accessing AWS Secrets Manager from Docker
Access to AWS Secrets Manager requires AWS credentials. Those
  credentials must have permissions to access the AWS resources that you
  want to access, such as your Secrets Manager secrets. The following
  sections provide details on how you can use AWS Identity and Access
  Management (IAM) policies to help secure access to your secrets and
  control who can access and administer them.


so you need to provide the aws credentials to docker which has access to the manger

you can follow on of these methods:

providing the credentials with run command:

docker run -e AWS_ACCESS_KEY_ID=XXXX -e AWS_SECRET_ACCESS_KEY=XXXX myimage


or providing the file ~/.aws/credentials and then create your container:

docker-machine create --driver amazonec2 --amazonec2-open-port 8000 --amazonec2-region us-west-1 aws-sandbox


see this