"Debian Support" option forces Artifactory to retrieve the 'Release' file from the remote repository, in case the package being fetched from the remote repository is newer than the the 'Release' file which is already cached. This is done in order to avoid the sort of errors you may encounter.
When you request a package, the client compares the checksum of the 'Packages' file with this file's checksum as it appears in the corresponding 'Release' file.
A mismatch can occur when you cached a package and the 'Release' file accordingly but this package was already modified (relative to the 'Release' file you have cached) when the next time you want to retrieve a package. This will cause the 'Packages' file to have a different checksum than the one that appears in the cached 'Release' file.
Enabling the 'Debian Support' for the remote repository will cause Artifactory to check if the 'Release' file is from the same time as the 'Packages' file (i.e. matches) on every request.
An exception to the description above is if the Debian client tries to first get the Release file and only after that retrieves the package. In this case, the client will get the "old" release .file from the cache of the remote repository and only then fetch the package from outside. In this case, running the 'update' command again should resolve the issue.