Find rsIDs for GRCh37 SNPs and rsIDs for GRCh38 SNPs and compare for overlap
it's not clear to me that it would support both versions of genome
  builds.


You can specify what version of the database you are querying.

I'd start with biomart on the ensembl website, see if you can get it to do what you want, then work on getting biomaRt to do what you want.