For almost everyone, a distro does not by it's self make it easier to backup.
From a marginal standpoint, one that has a smaller footprint would reduce network usage.
Do you want a full backup, some full with incremental to follow. Do you just need /home backed up or /data just backed up? We'd need to know more. Ftp is not secure also.
See rsync, dd over ftp or netcat, backula, clonezilla, and a number of distro's offer some version of backup. There are many more means too.
One might even create an automated mirror to drives by means of btrfs or zfs or mdadmin raid over iscsi or other iscsi channel mirror.
The options are really extensive.