You can pass the parameter BUILD_ID by using Predefined parameters option in the multi-job phase.
Steps are as follows:
Go to configure page, by clicking on Configure link of your multijob
Click on Advanced button of the job where you want to pass the parameter;
Click on Add Parameters, select Predefined parameters. Then create a parameter name of your own choice and assign the BUILD_ID to it;
Create the same parameter by enabling This build is parameterized option in downstream job where you want to receive the defined parameter. Then you can use the same in your script.