As Susan said, mounts can give valuable historic information about photographs. A crumbling mount material might also be problematic, though, and the best solution might be to contact a conservator or archivist who can carefully and safely remove the photographs from the mounts entirely, if the mounts are not of historical import. The American Institute for Conservation’s Find a Conservator tool can be useful: http://www.conservation-us.org/membership/find-a-conservator#.ViJ1zaQnhRl
If you decide to re-mount, a good material for re-mounting the photos depends a lot upon your storage, but an archival mat board and hinging material can easily be bought from a number of companies, including Hollinger, Talas, or Gaylord.
If the photographs are separated from their mounts, they will need to be stored differently, with interleaving in archival storage boxes.