Should we reinforce our laundry floor to prevent our washer from walking?
this is a common problem and is most easily remedied by one of two approaches (and this is entirely assuming the unit is in good working order and the harmonic balancer is in good shape and balance)

1)  restrain the unit - just mount angle iron or wood or whatever around the base to keep it from moving.  just ensure that you have a thick foam pad or strip around the base where the appliance "rubs" the restraint frame.  this will absorb some of the vibration and damp the transferred vibrational energy.

2)  replace the leveling feet with urethane elastomer vibration dampers (sorbothane or some such product - these are not just rubber feet).  they simply thread into the holes where the leveling feet were (just buy the right thread or some adapter to make it work).  they come in many different heights, styles, etc.  and though expensive, are far cheaper than reinforcing the floor or any such approach.  see here for details:

http://www.mcmaster.com/#standard-vibration-damping-mounts/=12buqho

fyi - i have done this for many customers machines and have even used a set on the washer we owned before our current unit - it fixed the problem essentially instantaneously