Here’s my method of cleaning bamboo blinds:

• First of all, I remove them from the windows. It’s tempting to skip this step, but if you don’t take them down, it’s almost impossible to clean them thoroughly on both sides, so your work won’t be as efficient as if you sink the blinds in a cleaning solution.
• I prepare a solution containing three parts water and one part laundry bleach, plus a few drops of vinegar
• With the blinds fully soaked in the solution, I wipe them thoroughly with a soft cloth, until all debris and mildew accumulations are gone.
• I then rinse the blinds very well with the shower, until the water runs clear.
• I let the blinds dry completely at air, and only then I hang them back at the windows.

As a special cautionary measure, I’d recommend you wear latex gloves when wiping the blinds, because the bleach could harm your skin.  A preventive fungicide treatment could help keep mildew off your bamboo blinds for a longer time, so doing it could be a good idea.