Write your parts without singers. Play vocal parts on an instrument such as a piano or guitar, and make sure that the harmonies you choose are part of the chord that is being played. If the band is playing in tune, everybody is sharing the same chord. Your backup vocals must do the same. Occasionally a note can go outside of the chord, but this should be a passing note on the way to a note that is in the chord.
Rehearse. Arranging backup vocals is not pure math. Magic can happen with actual singers in a room together. Use your written parts as a guide, but allow some leeway for backing vocalists to be creative. Listen for so-called mistakes that can actually make the vocals better, and capitalize on those mistakes by enhancing the new part rather than getting rid of it.
Leave space for the backup vocals in the recording. If the instruments are playing too many parts, there is no room for the vocals to contribute. This is called stepping on the track. During playback of the recording, if you hear a conflict between a backup vocal and an instrument, mute the track that instrument is on and make a note to check it or re-record it later.
Mix the backup vocals second. First set the lead vocal the way it sounds best, then bring in the backup vocals in the mix, with no instruments playing. This will support the lead vocal. Slowly bring in the other instruments, making sure the band also supports the lead singer. If your attention wanders to an instrument during a vocal section, you may have a track that has been overplayed.
Arrange your live show based on the recording. You don't need to get sloppy just because a show is live. Go back to the recording and make sure all of your musicians understand the role of supporting the singers. Often the live musicians will not be the same as the recording musicians, so rehearse the parts with an ear to making sure the backup vocals have a clear sonic space in the arrangement -- that is, make sure no one is playing a conflicting part on an instrument.